- Email: [email protected]
Starters and related codes
Journal of Statistical Planning and Inference 86 (2000) 379–395
www.elsevier.com/locate/jspi
Starters and related codes K. Chen, G. Ge, L. Zhu∗ Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China Received 28 March 1998; received in revised form 24 June 1998; accepted 30 October 1998
In Honor of Professor R.G. Stanton
Abstract Starters and their connections with codes are discussed. It is shown that there exists a skew starter in Zv for all v such that gcd(v; 6) = 1, v is either not divisible by 5 or divisible by 25. This can be used to show that there exists an optimal (6v; 4; 1) optical orthogonal code in Z6v for all v such that gcd(v; 6) = 1. By using frame starters with some special properties and their generalization, the spectrum of length n is determined to be n¿8 for maximum constant weight c 2000 Elsevier Science B.V. codes of distance 3 and weight 3 over an alphabet of size 7. All rights reserved. MSC: 05B07; 94B25 Keywords: Skew starter; Optical orthogonal code; Generalized Steiner triple system; Maximum constant weight code; Frame starter
1. Introduction and main results Let G be an additive abelian group of order v. A starter in G is a set of unordered pairs S = {{xi ; yi }: 16i6(v − 1)=2} which satisÿes the following two properties: (1) {xi : 16i6(v − 1)=2} ∪ {yi : 16i6(v − 1)=2} = G\{0}: (2) {±(xi − yi ): 16i6(v − 1)=2} = G\{0}. Clearly, there exists a starter in G only if v is odd. A starter S = {{xi ; yi }: 16i6(v − 1)=2} is said to be skew if it further satisÿes the following property: (3) {±(xi + yi ): 16i6(v − 1)=2} = G\{0}. Skew starters have been useful in constructions of Room squares and other combinatorial designs (see Dinitz and Stinson, 1992). Skew starters in Zv have special
Research supported in part by NSFC Grant 19831050. Corresponding author. E-mail address: [email protected] (L. Zhu) ∗
c 2000 Elsevier Science B.V. All rights reserved. 0378-3758/00/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 9 9 ) 0 0 1 1 9 - 6
380
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
application to constructions of Hamiltonian path balanced tournament designs (see Dinitz and Stinson, 1992). With the following known results, Dinitz and Stinson presented an open problem in Dinitz and Stinson (1992). Lemma 1.1 (Dinitz and Stinson, 1992). There is a skew starter in Zv for all v ¿ 5 if v is not divisible by 2; 3 and 5. There does not exist any skew starter in Zv if v ≡ 0 (mod 3). Open Problem (Dinitz and Stinson, 1992, Open Problem 2). Prove that there exists a skew starter in Zv for all v such that gcd(v; 6) = 1. In this article, we shall give a better answer than Lemma 1.1 to the open problem. Speciÿcally, we shall prove the following. Theorem 1.2. There exists a skew starter in Zv for all v such that gcd(v; 6) = 1; v is not divisible by 5 or v is divisible by 25. The results on skew starters in Zv can be used to discuss the existence of optical orthogonal codes. Let v; k and be positive integers. Following Brickell and Wei (1987), a (v; k; ) optical orthogonal code (OOC), C, is a family of (0; 1)-sequences of length v and weight k satisfying the following two properties: P (1) the auto-correlation property: 06t6v−1 xt xt+i 6, for any x =(x0 ; x1 ; : : : ; xv−1 ) ∈ C and any integer i 6≡ 0 (mod v); P (2) the cross-correlation property: 06t6v−1 xt yt+i 6, for any x 6= y in C and any integer i. Here, all subscripts are reduced modulo v. The study of OOCs is motivated by an application in a code-division multiple access ÿber optical channel which requires binary sequences with good correlation properties. For related details, the interested reader may refer to Brickell and Wei (1987), Chung et al. (1989), Salehi and Brackett (1989) and Salehi (1989). There are some combinatorial constructions for OOCs, especially for = 1, (see, for example, Bird and Keedwell, 1994; Yin, 1998). A (v; k; 1) OOC with |C| = b(v − 1)=k(k − 1)c codewords is said to be optimal. It was shown in Brickell and Wei (1987) that an optimal (v; 3; 1) OOC exists if and only if v 6= 6t + 2 and t ≡ 2 or 3 (mod 4). When k¿4, the existence problem for an optimal (v; k; 1) OOC remains unsolved. We shall use skew starters in Zv to construct optimal (6v; 4; 1) optical orthogonal code in Z6v and show the following. Theorem 1.3. There exists an optimal (6v; 4; 1) optical orthogonal code in Z6v for all v such that gcd(v; 6) = 1. In Section 3, we shall discuss some frame starters with special properties. This discussion leads to a class of maximum constant weight codes. A (g + 1)-ary constant weight code (n; w; d) is a code C ⊆(Zg+1 )n of length n and minimum distance d, such that every c ∈ C has Hamming weight w. To construct a constant weight code (n; w; d) with w = 3 a group divisible design (GDD) is used. A K − GDD(gn ) is an ordered
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
381
triple (P; G; B) where P is a set of gn elements, G is a collection of g-subsets called groups which partition P, and B is a set of some subsets called blocks of P, such that each block intersects each group in at most one element and that each pair of elements from distinct groups occurs together in exactly one block in B, where |B| ∈ K for any B ∈ B. In a 3 − GDD(gn ), let P = (Zg+1 \ {0}) × (Zn+1 \ {0}) with n groups Gi ∈ G; Gi = (Zg+1 \{0}) × {i}; 16i6n and blocks {(a; i); (b; j); (c; k)} ∈ B. One can construct a constant weight code (n; 3; d) as stated in Phelps and Yin (1997). From each block we form a codeword of length n by putting an a; b and c in positions i, j and k; respectively, and zeros elsewhere. This gives a constant weight code over Zg+1 with minimum distance 2 or 3. If the minimum distance is 3, then the code is a (g + 1)-ary maximum constant weight code (MCWC) (n; 3; 3) and the 3 − GDD(gn ) is called generalized Steiner triple system, denoted by GS(2; 3; n; g). The following is known. Lemma 1.4 (Etzion, 1997; Phelps and Yin, 1997). The necessary conditions for the existence of GS(2; 3; n; g) (or a (g + 1)-ary MCWC(n; 3; 3)) is that: (1) if g ≡ 0 (mod 6); then n¿g + 2; (2) if g ≡ 3 (mod 6); the n ≡ 1 (mod 2) and n¿g + 2; (3) if g ≡ 2 or 4 (mod 6); then n ≡ 0 or 1 (mod 3) and n¿g + 2; (4) if g ≡ 1 or 5 (mod 6); then n ≡ 1 or 3 (mod 6) and n¿g + 2. Lemma 1.5 (Etzion, 1997; Phelps and Yin, 1997; Phelps and Yin, to appear; Chen et al., to appear). There exists a (g + 1)-ary MCWC(n; 3; 3) for g = 2; 3; 4; 5 and 9 if and only if the necessary conditions in Lemma 1:4 are satisÿed with one exception of (g; n) = (2; 6). Recently, Blake-Wilson and Phelps (1999) proved that the necessary conditions listed in Lemma 1.4 are asymptotically sucient for any g. In this article, we shall show the following. Theorem 1.6. There exists a (g + 1)-ary MCWC(n; 3; 3) for g = 6 if and only if n¿8.
2. Skew starter and OOC In this section, we shall give the proof of Theorems 1.2 and 1.3. Lemma 2.1. There exists a skew starter in Z25 . Proof. We take S as follows: S = {{1; 2}; {3; 5}; {4; 7}; {6; 20}; {8; 13}; {9; 22}; {10; 17}; {11; 21}; {12; 18}; {14; 23}; {15; 19}; {16; 24}}: It is readily checked that S is a skew starter in Z25 .
382
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
Lemma 2.2. There exists a skew starter in Z125 . Proof. With the aid of a computer, we have found the following skew starter S in Z125 : S = {{1; 43}; {2; 49}; {3; 102}; {4; 116}; {5; 68}; {6; 117}; {7; 95}; {8; 60}; {9; 98}; {10; 11}; {12; 35}; {13; 67}; {14; 20}; {15; 70}; {16; 77}; {17; 122}; {18; 53}; {19; 115}; {21; 66}; {22; 61}; {23; 91}; {24; 45}; {25; 107}; {26; 103}; {27; 106}; {28; 38}; {29; 79}; {30; 58}; {31; 123}; {32; 83}; {33; 120}; {34; 92}; {36; 105}; {37; 52}; {39; 71}; {40; 124}; {41; 81}; {42; 118}; {44; 56}; {46; 55}; {47; 51}; {48; 108}; {50; 69}; {54; 84}; {57; 101}; {59; 112}; {62; 121}; {63; 74}; {64; 80}; {65; 82}; {72; 94}; {73; 78}; {75; 100}; {76; 110}; {85; 93}; {86; 104}; {87; 114}; {88; 119}; {89; 113}; {90; 97}; {96; 99}; {109; 111}}: Lemma 2.3. If there exist skew starters in Zn and Zm ; respectively; then there exists a skew starter in Znm . Proof. Let S1 = {{xi ; yi }: 16i6(n − 1)=2} and S2 = {{at ; bt }: 16t6(m − 1)=2} be skew starters in Zn and Zm , respectively. Let F = {{xi + jn; yi + 2jn}: 16i6(n − 1)=2; 06j6m − 1}; and S = F ∪ {{at n; bt n}: 16t6(m − 1)=2}: Notice that gcd(m; 6) = 1, it is easy to check that S is a skew starter in Znm . We can now prove Theorem 1.2. Proof of Theorem 1.2. If v is not divisible by 5, then the result comes from Lemma 1.1. If v is divisible by 25, then we can write v as v = 5k t, where k¿2 and gcd(t; 30) = 1. By Lemmas 2.1–2.3 we know that there exists a skew starter in Z5k . The existence of a skew starter in Zt is guaranteed by Lemma 1.1. So, again apply Lemma 2.3, we get the result. To answer the open problem mentioned in Section 1 completely, by Theorem 1.2, one needs only to consider the case v = 5t, where gcd(t; 30) = 1. In this case, the form v = 5p is fundamental where p ¿ 5 is an odd prime. For v ¡ 100 of this form, we have done a computer search and obtained the following. Lemma 2.4. There exists a skew starter in Zv for any v ∈ {35; 55; 65; 85; 95}. Proof. With the aid of a computer, we have found the following skew starters: v= 35; {1; 2}; {3; 5}; {4; 7}; {6; 10}; {8; 15}; {9; 21}; {11; 25}; {12; 29}; {13; 24}; {14; 30}; {16; 26}; {17; 22}; {18; 31}; {19; 34}; {20; 28}; {23; 32}; {27; 33}:
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
383
v= 55; {1; 3}; {2; 37}; {4; 40}; {5; 44}; {6; 46}; {7; 34}; {8; 9}; {10; 15}; {11; 18}; {12; 20}; {13; 43}; {14; 23}; {16; 26}; {17; 28}; {19; 45}; {21; 39}; {22; 35}; {24; 38}; {25; 49}; {27; 50}; {29; 41}; {30; 33}; {31; 52}; {32; 54}; {36; 53}; {42; 48}; {47; 51}: v= 65; {1; 42}; {2; 40}; {3; 23}; {4; 5}; {6; 10}; {7; 62}; {8; 44}; {9; 32}; {11; 51}; {12; 58}; {13; 45}; {14; 16}; {15; 18}; {17; 38}; {19; 31}; {20; 25}; {21; 52}; {22; 61}; {24; 35}; {26; 41}; {27; 49}; {28; 56}; {29; 64}; {30; 36}; {33; 46}; {34; 48}; {37; 53}; {39; 57}; {43; 60}; {47; 54}; {50; 59}; {55; 63}: v= 85; {1; 24}; {2; 72}; {3; 62}; {4; 40}; {5; 78}; {6; 73}; {7; 47}; {8; 42}; {9; 39}; {10; 57}; {11; 71}; {12; 68}; {13; 63}; {14; 15}; {16; 84}; {17; 38}; {18; 50}; {19; 43}; {20; 26}; {21; 52}; {22; 31}; {23; 66}; {25; 27}; {28; 36}; {29; 49}; {30; 69}; {32; 80}; {33; 74}; {34; 41}; {35; 51}; {37; 64}; {44; 77}; {45; 59}; {46; 65}; {48; 61}; {53; 75}; {54; 82}; {55; 58}; {56; 67}; {60; 70}; {76; 81}; {79; 83}: v= 95; {1; 37}; {2; 77}; {3; 61}; {4; 82}; {5; 72}; {6; 67}; {7; 34}; {8; 43}; {9; 17}; {10; 60}; {11; 85}; {12; 50}; {13; 26}; {14; 44}; {15; 84}; {16; 71}; {18; 94}; {19; 63}; {20; 69}; {21; 54}; {22; 24}; {23; 79}; {25; 89}; {27; 33}; {28; 46}; {29; 32}; {30; 62}; {31; 74}; {35; 49}; {36; 45}; {38; 42}; {39; 68}; {40; 87}; {41; 52}; {47; 88}; {48; 90}; {51; 73}; {53; 78}; {55; 70}; {56; 66}; {57; 80}; {58; 65}; {59; 83}; {64; 76}; {75; 91}; {81; 86}; {92; 93}: At present, we are not able to determine the existence of a skew starter of this case completely. However, we may construct some optimal (6v; 4; 1) OOC by using the known results obtained above. We need the concept of cyclic packing (see Yin, 1998). Let B = {B1 ; B2 ; : : : ; Bt }, where Bi = {bi1 ; bi2 ; : : : ; bik }, bij ∈ Zv , 16i6t and 16j6k. Let D = {bij − bis : 16i6t; j 6= s; 16j; s6k}. The pair (Zv ; B) is said to be a cyclic packing CP(v; k; 1) if the cardinality of D is k(k − 1)t and 0 6∈ D. Further, it is called optimal when t = b(v − 1)=k(k − 1)c. The following result was presented in Yin (1998). Lemma 2.5 (Yin, 1998). The existence of an optimal (v; k; 1) OOC is equivalent to the existence of an optimal CP(v; k; 1). Lemma 2.6. If there exists a skew starter in Zv such that gcd(v; 6) = 1; then there exists an optimal (6v; 4; 1) OOC. Proof. Let S = {{xi ; yi }: 16i6(v − 1)=2} be a skew starter in Zv , where v = 2t + 1. In Zv × Z6 , which is isomorphic to Z6v since gcd(v; 6) = 1, let Bi = {(xi ; 0); (yi ; 0); (xi + yi ; 1); (0; 4)}, 16i6t. We show that the 12t di erences provided by Bi are nonzero and di erent from each other. These di erences can be divided into 6 classes according to the second coordinate: (1) ±(xi − yi ; 0), 16i6t, (2) (yi ; 1), (xi ; 1), 16i6t,
384
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
(3) (xi ; 2), (yi ; 2), 16i6t, (4) (xi + yi ; 3), (−xi − yi ; 3), 16i6t, (5) (−xi ; 4), (−yi ; 4), 16i6t, (6) (−yi ; 5), (−xi ; 5), 16i6t. By the deÿnition of S, it is easily seen that the 2t di erences in each of the classes (1) – (6) are nonzero and di erent from each other as expected. It follows that Bi (16i6t) form a CP(6v; 4; 1) in Zv ×Z6 . Since b(6v −1)=4(4−1)c =b(12t +5)=12)c =t, so the cyclic packing is optimal. By Lemma 2.5, there exists an optimal (6v; 4; 1) OOC. Lemma 2.7. There exists an optimal CP(6v; 4; 1) for all v = 5h; h ¿ 1 and gcd(h; 30) = 1. Proof. Since gcd(h; 30) = 1, by Lemma 1:2, there exists a skew starter in Zh : S = {{xi ; yi }: 16i6(h − 1)=2}: Let F = {{x + jh; y + 2jh}: {x; y} ∈ S; 06j64}; in Zv and let B = {{( ; 0); ( ; 0); ( + ; 1); (0; 4)}: { ; } ∈ F}; A = {{(0; 0); (h; 1); (3h; 3); (0; 4)}; {(0; 0); (0; 5); (3h; 1); (4h; 1)}}; in Zv × Z6 . Then B ∪ A is an optimal CP(6v; 4; 1) in cyclic group Zv × Z6 . In fact, the 4-subsets in B provide 12 × 5 × (h − 1)=2 di erences which form the set (Z5h\{0; h; 2h; 3h; 4h})×Z6 . Meanwhile, the 4-subsets in A provide 24 di erences which form the set ({0; h; 2h; 3h; 4h} × Z6 )\{(0; 0); (0; 3); ±(h; 2); ±(2h; 0)}. So the result is obtained. We can now prove Theorem 1.3. Proof of Theorem 1.3. If v = 5t; t ¿ 1 and gcd(t; 5) = 1, then the result comes from Lemmas 2.7 and 2.5. Otherwise, v is not divisible by 5 or is divisible by 25, there exists a skew starter in Zv by Theorem 1.2. So we get the result by Lemma 2.6. 3. Frame starters and MCWC An important generalization of a starter is the frame starter. In this section we shall use frame starters with some additional properties ∗ FS(2n ) and their variations ∗ GFS(2n ) to construct GS(2; 3; n; 6) for small values of n. We shall also apply PBD construction to prove Theorem 1.6. Let G be an additive abelian group of order nh and H be a subgroup of order h, where nh − h is even. A frame starter in G\H of type hn (simply, FS(hn )) is a set of
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
385
unordered pairs S = {{xi ; yi }: 16i6(nh − h)=2} such that the following two properties are satisÿed: (1) {xi : 16i6(nh − h)=2} ∪ {yi : 16i6(nh − h)=2} = G\H . (2) {±(xi − yi ): 16i6(nh − h)=2} = G\H . Thus, a starter is the special case of a frame starter when H = {0}. From an FS(hn ) one may easily get a 3−GDD((3h)n ) (P; G; B) as follows: P = G × Z3 ; G = {G1 ; G2 ; : : : ; Gn }; Gj = Hj × Z3 ; Hj is the jth coset of H in G, and Bi = {(xi ; 0); (yi ; 0); S (0; 1)}; 16i6(nh − h)=2, B = 16i6(nh−h)=2 {Bi + (a; b): a ∈ G; b ∈ Z3 }. It is clear that the corresponding constant weight code contains two codewords having distance 2 i the 3−GDD contains two intersecting blocks which intersect only three groups of the 3−GDD. By the above construction this can happen only if the blocks {(0; 0); (yi − xi ; 0); (−xi ; 1)}; {(xi − yi ; 0); (0; 0); (−yi ; 1)}; {(xi ; 2); (yi ; 2); (0; 0)}; 16i6(nh − h)=2; contain such a pair of blocks. This means that the pairs {hyi − xi in ; h−xi in }; {hxi − yi in ; h−yi in }; {hxi in ; hyi in } for 16i6(nh − h)=2 cannot be all distinct, where hxin = j when x ∈ Gj . Speciÿcally, hxin ≡ x (mod n) when x ∈ G = Zhn , and h(x; t)in = x when (x; t) ∈ G = Zn × Zh . If these pairs are all distinct, then we obtain a GS(2; 3; n; 3h). Denote such an FS(hn ) by ∗ FS(hn ). Thus, we have proved the following: Lemma 3.1. If there exists a
∗
FS(hn ); then there exists a GS(2; 3; n; 3h).
In the remainder of this article we shall focus on the existence of a GS(2; 3; n; 6). From Lemma 3.1, we need to discuss the existence of ∗ FS(2n ). Lemma 3.2. There exists a
∗
FS(28 ) in Z16 \{0; 8}.
Proof. Let S = {{1; 2}; {3; 5}; {4; 10}; {6; 13}; {7; 11}; {9; 14}; {12; 15}}: It is readily checked that S is a ∗ FS(28 ) in Z16 \{0; 8}. Let [a; b] denote the set of integers in the interval a6n6b. We shall discuss the method of PBD-closure which reduces the existence of a GS(2; 3; n; 6) to the set E = [8; 79]\{57; 64; 65; 72; 73, 74}. We are quite successful in ÿnding a ∗ FS(2n ) for most values n ∈ E1 = {v ∈ E: v ≡ 0; 1(mod 4)}, excepting for n = 9. To ÿnd a ∗ FS(2n ) in Z2n \{0; n} for the remaining values n of E1 , we can take a suitable multiplier m ∈ Z2n \{0; n} and a suitable integer t, and then try to ÿnd two St−1 sets of pairs, P0 and R, so that S = P ∪ R forms a ∗ FS(2n ), where P = i=0 Pi and Pi = mi P0 ; 16i6t − 1. We call such a set P0 partial set and R remainder. Lemma 3.3. There exists a
∗
FS(2n ) in Z2n \{0; n} for all n ∈ {12; 16; 20}.
386
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
Proof. With the aid of a computer, we have found a ∗ FS(2n ) in Z2n \ {0; n} for all n ∈ {12; 16; 20}. As mentioned above, for a ∗ FS(2n ), we only list the corresponding multiplier m, integer t, partial set P0 and remainder R as follows. ∗ FS(212 ); m = 3; t = 2; P0 = {{1; 2}}; R = {{4; 9}; {5; 11}; {7; 14}; {8; 19}; {10; 20}; {13; 22}; {15; 23}; {16; 18}; {17; 21}}: ∗ FS(216 ); m = 3; t = 4; P0 = {{1; 2}}; R = {{8; 20}; {10; 23}; {12; 30}; {14; 21}; {17; 28}; {7; 24}; {5; 31}; {13; 15}; {4; 26}; {11; 19}; {25; 29}}: ∗ FS(220 ); m = 3; t = 4; P0 = {{1; 2}; {4; 11}}; R = {{21; 25}; {5; 30}; {7; 29}; {8; 13}; {26; 34}; {10; 16}; {24; 38}; {22; 32}; {15; 31}; {23; 35}; {37; 39}}: Now, we discuss the method of PBD-closure. A pairwise balanced design of order v ((v; K)-PBD) is an ordered pair (X; B), where X is a set (of points) of size v and B is a collection of subsets of X (called blocks) with the property that every pair of elements of X occurs together in exactly one block B ∈ B, |B| ∈ K. Let B(K) denote the PBD-closure of K, i.e., the set of positive integers n for which there exists a PBD(n; K). K is PBD-closed if B(K) = K. It is well known (see Beth et al., 1986) that if K is a closed set, then there exists a ÿnite subset J ⊆ K such that K = B(J ). This set J is called a generating set for the PBD-closed set K. Let T = {n: there exists a GS(2; 3; n; 6)}. By the deÿnition of a GS(2; 3; n; g) and the well-known Wilson’s fundamental construction for GDDs (see Colbourn and Dinitz, 1996, III Theorem 2.5), it is easy to see that T is PBD-closed. Let K8 be the set of positive integers n¿8. By the necessary conditions stated in Lemma 1.4, we have T ⊆ K8 . To prove Theorem 1.6, it suces to prove that K8 ⊆ T . To do this, we need only to show that E is a generating set for K8 and that E ⊆ T . Lemma 3.4 (Colbourn and Dinitz, 1996). K8 =B(E); where E =[8; 79]\{57; 64; 65; 72; 73; 74}. Let E2 = E \ E1 . That is, E2 = {v ∈ E: v ≡ 2; 3 (mod 4)}. It is not dicult to see that there is no FS(2n ), consequently, no ∗ FS(2n ) in Z2n \{0; n} for n ≡ 2 or 3 (mod 4) (see Dinitz and Stinson, 1992, Theorem 2:14). So, the method given in Lemma 3.1 is failed to give a GS(2; 3; n; 6) for any n ∈ E2 . We need a generalized concept of FS(hn ) and ∗ FS(hn ). Let H be a subgroup of order h of Znh , where nh − h is even. Let w be a positive integer such that w|nh. A generalized frame starter in Znh \ H , with respect to w, Sw−1 of type (hn ) (simply, GFS(hn ; w)), is a set of unordered pairs S = j=0 Sj , where Sj = {{xij ; yij }: 16i6(nh − h)=2}; 06j6w − 1, such that the following two properties are satisÿed: (1) {xij − j: 16i6(nh − h)=2} ∪ {yij − j: 16i6(nh − h)=2} = Znh \H; 06j6w − 1, Sw−1 (2) j=0 ({(xij −yij ; hxij iw ): 16i6(nh−h)=2}∪{(yij −xij ; hyij iw ): 16i6(nh−h)=2})= (Znh \H ) × Zw . Let aij =hxij −jin ; bij =hyij −jin , and cij =hyij −xij in for 16i6(nh−h)=2, 06j6w−1. S is called a ∗ GFS(hn ; w) if it further satisÿes the following property that in Zn ×Zn ×Zw ,
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
387
all of the following 3w(nh − h) ordered triples (aij ; bij ; hjiw ); (bij ; aij ; hjiw ); (cij ; n − aij ; hxij iw ); (n − aij ; cij ; hxij iw ); (n − cij ; n − bij ; hyij iw ); (n − bij ; n − cij ; hyij iw ) for 16i6(nh − h)=2; 06j6w − 1, are distinct. Obviously, a GFS(hn ; 1) (or ∗ GFS(hn ; 1)) is just an FS(hn ) (or ∗ FS(hn )). For any w such that w|nh, an FS(hn ) (or ∗ FS(hn )) is also a GFS(hn ; w) (or ∗ GFS(hn ; w)). We have a construction for a GS(2; 3; n; 3h) which is similar to that of Lemma 3.1. Lemma 3.5. If there exists a ∗ GFS(hn ; w) for some positive integer w such that w|nh; then there exists a GS(2; 3; n; 3h). Sw−1 Proof. Let S = j=0 Sj be a ∗ GFS(hn ; w) in Znh \ H , where Sj = {{xij ; yij }: 16i6 (nh − h)=2}; {06j6w − 1}. Let Bij = {(xij ; 0); (yij ; 0); (j; 1)}; 16i6(nh − h)=2; { 06j6w − 1}; and B=
(nh−h)=2 w−1 [ [ i=1
{Bij + (aw; b): 16a6nh=w; b ∈ Z3 }:
j=0
We shall show that (Znh × Z3 ; G; B) is a 3−GDD((3h)n ), where G = {G0 ; G1 ; : : : ; Gn−1 }, and Gi = (H + i) × Z3 (06i6n − 1) are groups. In fact, |B| =3h2 n(n − 1)=2. So, it suces to show that for any two groups Gs and Gt , s 6= t, and for any ( ; ) ∈ Gs ; ( ; ) ∈ Gt , there exists a block A ∈ B such that ( ; ) and ( ; ) lie in A together. We shall prove this in two cases. (i) If = , then by the deÿnition of a GFS(hn ), we know that there must exist a pair {x; y} ∈ Sj for some j; 06j6w − 1, such that (x − y; hxiw ) = ( − ; h iw ): Since hxiw = h iw , we have − x = − y = aw for some a ∈ Znh . This forces (y − x; hyiw ) = ( − ; h iw ): Take A = {(x; 0); (y; 0); (j; 1)} + (aw; ). Obviously, A ∈ B and ( ; ) and ( ; ) lie in A. (ii) If 6= , without loss of generality, we may assume that − = 1 in Z3 . Let j = h iw ; then − j = aw for some a ∈ Znh . Since ( ; ) ∈ Gs ; ( ; ) ∈ Gt ; s 6= t, we have − ∈ Znh \H . So, − j − aw ∈ Znh \H . By the deÿnition of a GFS(hn ), there exists a pair {x; y} ∈ Sj such that x − j = − j − aw, or y − j = − j − aw, i.e. x (or y)= − aw in Znh . Take A = {(x; 0); (y; 0); (j; 1)} + (aw; ) in B. Clearly, ( ; ) and ( ; ) lie in A. Next we show that the 3−GDD is a GS(2; 3; n; 3h). Otherwise, the 3−GDD contains two blocks which have a common point and intersect three groups only. Suppose
388
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
that the two blocks are {(x; i); (y; j); (z; k)} and {(x; i); (y; j0 ); (z; k 0 )}, where y ≡ y (mod n); z ≡ z (mod n). Without loss of generality, we have in the ∗ GFS(hn ; w) two pairs {xpq ; ypq } ∈ Sq and {xst ; yst } ∈ St such that x = xpq + aw; y = xpq + aw; z = q + aw; {x; y; z} = {xst + cw; yst + cw; t + cw} for some a; c ∈ Znh . We consider only one special case here, and the other possibilities are similarly proved. Suppose x = xst + cw;
y = yst + cw;
z = t + cw:
Thus, we have ypq − xpq = y − x ≡ y − x = yst − xst (mod n); q − xpq = z − x ≡ z − x = t − xst (mod n); xpq ≡ xst (mod w): This means that the two ordered triples (hypq − xpq in ; hq − xpq in ; hxpq iw ) and (hyst − xst in ; ht − xst in ; hxst iw ) are the same. This contradicts the deÿnition of a ∗ GFS(hn ). ∗
From this, we know that to construct a GS(2; 3; n; 6) for n ∈ E2 , it suces to ÿnd a GFS(2n ; w) for some positive integer w such that w|2n. We have the following.
Lemma 3.6. There exists a
∗
GFS (29 ; 3) in Z18 \{0; 9}.
Proof. With the aid of a computer, a ∗ GFS(29 ; 3) in Z18\{0; 9} has been found, which is listed as follows: S0 = {{16; 12}; {1; 3}; {15; 2}; {7; 13}; {14; 6}; {17; 5}; {10; 11}; {8; 4}}; S1 = {{11; 4}; {5; 13}; {2; 3}; {16; 6}; {14; 17}; {15; 9}; {7; 0}; {8; 12}}; S2 = {{17; 6}; {15; 16}; {13; 0}; {9; 12}; {4; 7}; {5; 3}; {1; 14}; {10; 8}}: Lemma 3.7. For any n ∈ F = {10; 11; 14; 15; 18; 19; 23; 26; 35; 38; 47}; there exists a ∗ GFS (2n ; w) in Z2n \{0; n} for some positive integer w such that w|2n. Proof. For each n ∈ F, with the aid of a computer, we have found a ∗ GFS(2n ; w) for some positive integer w such that w|2n. Here we only list the ∗ GFS(2n ; w)s for n ∈ {10; 11; 14; 15}. For other values in F, we list the corresponding ∗ GFS(2n ; w)s in the appendix. ∗ GFS(210 ; 4): S0 = {{12; 19}; {4; 7}; {9; 6}; {13; 2}; {15; 16}; {18; 11}; {3; 14}; {1; 17}; {5; 8}}; S1 = {{16; 10}; {5; 12}; {3; 9}; {13; 18}; {0; 4}; {2; 14}; {7; 19}; {6; 8}; {17; 15}}; S2 = {{7; 1}; {3; 6}; {4; 13}; {0; 11}; {17; 9}; {16; 8}; {18; 5}; {14; 10}; {15; 19}}; S3 = {{9; 4}; {8; 10}; {16; 11}; {2; 7}; {17; 19}; {5; 6}; {18; 12}; {14; 15}; {0; 1}}: ∗ GFS(211 ; 2): S0 = {{7; 21}; {15; 14}; {17; 12}; {8; 5}; {20; 10}; {2; 18}; {16; 1}; {19; 6}; {3; 9}; {13; 4}};
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
389
S1 = {{2; 17}; {19; 16}; {10; 18}; {14; 9}; {5; 6}; {11; 21}; {13; 15}; {0; 20}; {3; 7}; {4; 8}}: ∗ GFS(214 ; 4): S0 = {{4; 5}; {1; 6}; {23; 16}; {21; 18}; {19; 10}; {22; 27}; {26; 13}; {15; 2}; {11; 7}; {24; 8}; {12; 20}; {25; 17}; {9; 3}}; S1 = {{23; 5}; {21; 17}; {19; 16}; {8; 14}; {22; 11}; {18; 25}; {27; 6}; {2; 12}; {9; 10}; {0; 13}; {24; 26}; {3; 4}; {7; 20}}: S2 = {{15; 20}; {22; 5}; {26; 10}; {17; 23}; {1; 24}; {19; 7}; {0; 11}; {14; 18}; {6; 3}; {8; 12}; {25; 27}; {21; 9}; {4; 13}}; S3 = {{21; 2}; {6; 26}; {14; 4}; {5; 8}; {1; 27}; {16; 9}; {24; 13}; {20; 11}; {10; 12}; {18; 19}; {25; 7}; {15; 23}; {22; 0}}: ∗ GFS(215 ; 2): S0 = {{25; 18}; {7; 29}; {13; 8}; {14; 2}; {21; 17}; {27; 10}; {28; 22}; {19; 9}; {11; 20}; {23; 26}; {1; 12}; {6; 4}; {5; 3}; {16; 24}}; S1 = {{9; 28}; {13; 27}; {15; 3}; {23; 10}; {0; 14}; {7; 4}; {24; 19}; {21; 20}; {6; 29}; {2; 12}; {11; 5}; {17; 8}; {25; 26}; {18; 22}}: For the remaining values n in E, we can also ÿnd a ∗ FS(2n ) when n ∈ E1 or a GFS(2n ; w) when n ∈ E2 , consequently getting a GS(2; 3; n; 6). We had done so in the ÿrst version of this paper. To save space we now deal with these values in other ways, which are described below. It is natural to consider the possibility of using Wilson’s fundamental construction for GDDs to construct GS(2; 3; n; g)s. This construction is also called weighting construction, see (Beth et al., 1986, Theorem IX. 3.2) or (Colbourn and Dinitz, 1996, Theorem III. 2.5). However, the resulting 3−GDD may not have the property that any two intersecting blocks intersect at most two common groups even though the ingredient 3−GDDs do. The reason is that the master GDD may have two intersecting blocks which intersect more than two common groups. In this case, the two ingredient 3−GDDs may contain a block from each one violating the required property. This analysis suggests that we may enforce conditions on the master GDD to avoid such things to happen. This leads to the following deÿnition. Deÿnition 3.1. A K−GDD is said to have “star” property and denoted by K−∗ GDD if any two intersecting blocks intersect at most two common groups. With this deÿnition a GS(2; 3; n; g) is just the same as a 3−∗ GDD(gn ). Using a K−∗ GDD as a master GDD, the well-known Wilson’s fundamental construction can be used to construct GS(2; 3; n; g)s, which we state below. Lemma 3.8 (Weighting). Let (V; G; B) be a K −∗ GDD (the master GDD) with groups G1 ; G2 ; : : : ; Gt . Suppose there exists a function w: V → Z + ∪ {0} (a weighting function) which has the property that for each block B ={x1 ; x2 ; : : : ; xk } ∈ B there exists a 3−∗ GDD of group type (w(x1 ); w(x2 ); : : : ; w(xk )) (such a GDD is an “ingredient”
390
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
P
GDD). Then there exists a 3 −∗ GDD of group type ( P x∈Gt w(x)).
x∈G1
w(x);
P x∈G2
w(x); : : : ;
Proof. The original Wilson’s fundamental construction guarantees that the resultant design is a 3 − GDD. To see the star property we take two blocks from the resultant 3 − GDDs. If they belong to di erent ingredient 3 − GDDs, they must have the star property since the master K−∗ GDD does. Otherwise, they belong to the same ingredient 3 −∗ GDD. So, they must also have the star property. As a corollary the following lemma is useful. Lemma 3.9. If there exists a K −∗ GDD(gn ); and there exists a GS(2; 3; k; h) for any k ∈ K; then there exists a GS(2; 3; n; gh). Proof. In Lemma 3.8, take the K −∗ GDD(gn ) as the master GDD and the GS(2; 3; k; h)s as ingredient 3 −∗ GDDs, the weighting function is w(x) = h for any element x in the master GDD. Lemma 3.10. There exists a 4 −∗ GDD(3n ) for n ∈ {13; 17; 21}. Proof. With the aid of a computer, we have found a set of base blocks A for such a 4 −∗ GDD(3n ) in Z3n with the groups Gi = {i; n + i; 2n + i}, 06i6n − 1, which is listed as follows. n = 13; A = {{0; 1; 6; 31}; {0; 2; 12; 23}; {0; 3; 7; 22}}: n = 17; A = {{0; 1; 3; 8}; {0; 4; 18; 30}; {0; 6; 19; 29}; {0; 9; 20; 36}}: n = 21; A = {{0; 1; 3; 7}; {0; 5; 13; 35}; {0; 9; 20; 38}; {0; 10; 24; 47}; {0; 12; 31; 48}}. By Lemma 1.5 we know that there exists a GS(2; 3; 4; 2). So, by Lemmas 3.10 and 3.9 we have the following. Lemma 3.11. There exists a GS(2; 3; n; 6) for n ∈ {13; 17; 21}. So far, we have proven the following. Lemma 3.12. There exists a GS(2; 3; n; 6) for n ∈ [8; 21] ∪ {23; 26; 35; 38; 47}. Etzion in 1997 established some product constructions for GS(2; 3; n; g)s, some of which can be stated as follows. Lemma 3.13 (Etzion, 1997, Constructions C and D). Let (V; G; B) be a 3 − GDD (gm ). Suppose there exists a GS(2; 3; n; g). Then there exists a GS(2; 3; mn; g) or a GS(2; 3; m(n − 1) + 1; g) if B can be partitioned into t sets S0 ; S1 ; : : : ; St−1 ; such that the minimum distance in Sr , 06r6t − 1; is 3; and t6n or t6n − 1; respectively.
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
391
Lemma 3.14. There exists a 3 − GDD(63 ) with the property that all blocks of the design can be partitioned into t sets S0 ; S1 ; : : : ; St−1 such that t67 and the minimum distance in Sr ; 06r6t − 1, is 3. Proof. Let V = Z18 , G = {G0 ; G1 ; G2 }, where Gi = {i + 3j: 06j65}, 06i62, and S6 B = r=0 Sr , where S0 = {{1; 8; 12}; {7; 14; 0}; {13; 2; 6}; {3; 10; 17}; {9; 16; 5}; {15; 4; 11}}, S1 = {{1; 9; 11}; {7; 15; 17}; {13; 3; 5}; {0; 2; 10}; {6; 8; 16}; {12; 14; 4}}, S2 = {{2; 4; 9}; {8; 10; 15}; {14; 16; 3}; {0; 1; 17}; {6; 7; 5}; {12; 13; 11}}, S3 = {{1; 2; 3}; {7; 8; 9}; {13; 14; 15}; {0; 4; 5}; {6; 10; 11}; {12; 16; 17}}, S4 = {{9; 10; 14}; {7; 11; 3}; {13; 0; 8}; {4; 6; 17}}, S5 = {{15; 16; 2}; {13; 17; 9}; {1; 6; 14}; {10; 12; 5}}, S6 = {{3; 4; 8}; {1; 5; 15}; {7; 12; 2}; {16; 0; 11}}. It is easy to check that (V; G; B) is a 3 − GDD(63 ) and the minimum distance in Sr , 06r66, is 3. So, t67. Lemma 3.15. There exists a 3 − GDD(6m ) for m = 4; 5 with the property that all blocks of such a design can be partitioned into t sets S0 ; S1 ; : : : ; St−1 such that t64 and the minimum distance in Sr , 06r6t − 1; is 3. Proof. By Lemma 1.5 there exists a GS(2; 3; 4; 2) and a GS(2; 3; 5; 3). Use these designs as master GDDs in Wilson’s Fundamental construction and give weight three or two to each element, we get a 3 − GDD(64 ) or a 3 − GDD(65 ), respectively. It is not dicult to see that the resultant designs are desired ones. By Lemmas 3.13–3.15 we have the following. Lemma 3.16. If there exists a GS(2; 3; n; 6); then there exists a GS(2; 3; mn; 6) and a GS(2; 3; m(n − 1) + 1; 6); where m = 3; 4; 5. Proof. Since a GS(2; 3; n; 6) exists, we have from Lemma 1.4 that n¿8. For m=3; 4; 5, a 3 − GDD(6m ) exists from Lemmas 3.14 –3.15 with t67. Then the conclusion comes from Lemma 3.13. Lemma 3.17. There exists a GS(2; 3; v; 6) for v ∈ E\([8; 21]∪{23; 26; 35; 38; 47; 59; 62}). Proof. For any v ∈ E \([8; 21] ∪ {23; 26; 35; 38; 47; 59; 62}), we can write v = mn or v = m(n − 1) + 1 for some m ∈ {3; 4; 5} and n ∈ [8; 21] ∪ {23; 26}. By Lemmas 3.12 and 3.16 there exists a GS(2; 3; v; 6). Here, we list the triples (v; m; n) in Table 1. To determine the existence of GS(2; 3; n; 6) for the remaining values n of E, singular indirect product (SIP) construction for GS(2; 3; n; g)s (see Chen et al. (1999), Theorem 2:3) will be used. We need some notations. A holey group divisible design, K −HGDD, is a four-tuple (V; G; H; B), where V is a set of points, G is a partition of V into
392
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
Table 1 Triples (v; m; n) for v ∈ E \([8; 21] ∪ {23; 26; 35; 38; 47; 59; 62}) v
m
n
v
m
n
v
m
n
22 = 3 · 7 + 1 27 = 3 · 9 30 = 3 · 10 33 = 3 · 11 37 = 3 · 12 + 1 41 = 4 · 10 + 1 44 = 4 · 11 48 = 3 · 16 51 = 3 · 17 54 = 3 · 18 58 = 3 · 19 + 1 63 = 3 · 21 68 = 4 · 17 71 = 5 · 14 + 1 77 = 4 · 19 + 1
3 3 3 3 3 4 4 3 3 3 3 3 4 5 4
8 9 10 11 13 11 11 16 17 18 20 21 17 15 20
24 = 3 · 8 28 = 3 · 9 + 1 31 = 3 · 10 + 1 34 = 3 · 11 + 1 39 = 3 · 13 42 = 3 · 14 45 = 3 · 15 49 = 3 · 16 + 1 52 = 3 · 17 + 1 55 = 3 · 18 + 1 60 = 3 · 20 66 = 5 · 15 + 1 69 = 3 · 23 75 = 5 · 15 78 = 3 · 26
3 3 3 3 3 3 3 3 3 3 3 5 3 5 3
8 10 11 12 13 14 15 17 18 19 20 16 23 15 26
25 = 3 · 8 + 1 29 = 4 · 7 + 1 32 = 4 · 8 36 = 3 · 12 40 = 3 · 13 + 1 43 = 3 · 14 + 1 46 = 3 · 15 + 1 50 = 5 · 10 53 = 4 · 13 + 1 56 = 4 · 14 61 = 3 · 20 + 1 67 = 3 · 22 + 1 70 = 5 · 14 76 = 4 · 19 79 = 3 · 26 + 1
3 4 4 3 3 3 3 5 4 4 3 3 5 4 3
9 8 8 2 14 15 16 10 14 14 21 23 14 19 27
subsets called groups, H ⊂ G; B is a set of blocks such that a group and a block contain at most one common point and every pair of points from distinct groups, not both in H, occurs in a unique block in B, where |B| ∈ K for any B ∈ B. A k − HGDD(g(n;u) ) denotes a K − HGDD with n groups of size g in G, u groups in H and K = {k}. A holey generalized Steiner triple system, HGS(2; 3; (n; u); g), is a 3 − HGDD(g(n;u) ) with the property that any two intersecting blocks intersect at most two common groups. It is easy to see that if u = 0 or u = 1, then a HGS(2; 3; (n + u; u); g) is just a GS(2; 3; n; g) or a GS(2; 3; n + 1; g), respectively. The following lemma is clear. Lemma 3.18 (Filling in holes). If there exists a HGS(2; 3; (n; u); g) and a GS(2; 3; u; g); then there exists a GS(2; 3; n; g). Take m = 3 and g = 6 in Theorem 2:3 in Chen et al. (1999). Let n¿7 be a prime power, then the desings (1) and (2) stated in Theorem 2:3 in Chen et al. (1999) are guaranteed by Lemma 3.2 in Chen et al. and Lemma 3.14. So, we have the following. Lemma 3.19. Let n¿7 be a prime power and u; a be integers such that 06a6u6n. Suppose the following designs exist: (1) a HGS(2; 3; (n + u; u); 6); (2) a GS(2; 3; 2a + u; 6). Then there exists a GS(2; 3; 3n + 2a + u; 6). Lemma 3.20. There exists a GS(2; 3; n; 6) for n ∈ {59; 62}. Proof. A GS(2; 3; 25; 6) with a sub GS(2; 3; 9; 6) is given in the proof of Lemma 3.17. Delete the blocks of this sub GS(2; 3; 9; 6) from the GS(2; 3; 25; 6) to get a
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
393
HGS(2; 3; (25; 9); 6). A GS(2; 3; 11; 6) is guaranteed by Lemma 3.12. By taking n = 16; u = 9 and a = 1 in Lemma 3.19, a GS(2; 3; 59; 6) is obtained. Similarly, the proof of Lemma 3.17 provides a GS(2; 3; 24; 6) with a sub GS(2; 3; 8; 6). As is done above, a GS(2; 3; (24; 8); 6) is obtained. The existence of a GS(2; 3; 14; 6) is guaranteed by Lemma 3.12. Take n = 16; u = 8 and a = 3 in Lemma 3.19 to get a GS(2; 3; 62; 6). We are now in a position to prove Theorem 1.6. Proof of Theorem 1.6. The necessity comes from Lemma 1.4. For suciency, by Lemmas 3.12, 3.17 and 3.20 we know that there exists a GS(2; 3; n; 6) for any n ∈ E. So, there exists a GS(2; 3; n; 6) for any n¿8. Consequently, there exists a (6 + 1)-ary MCWC (n; 3; 3) for any n¿8. Note added in proof (June 1999) The existence of a GS(2,3,n; g) has been solved also for g = 7; 8 in Wu et al. (1999) by di erent methods. Acknowledgements The authors would like to thank the referees for their helpful comments. Appendix A. ∗
GFS(218 ; 4): S0 = {{24; 13}; {7; 1}; {17; 20}; {5; 4}; {15; 27}; {26; 19}; {23; 31}; {28; 2}; {12; 29}; {30; 21}; {14; 6}; {35; 10}; {34; 32}; {16; 11}; {33; 25}; {22; 8}; {3; 9}}; S1 = {{34; 0}; {6; 29}; {10; 11}; {28; 4}; {31; 21}; {14; 23}; {9; 33}; {18; 24}; {17; 7}; {8; 12}; {5; 27}; {32; 2}; {22; 26}; {15; 16}; {20; 13}; {35; 3}; {25; 30}}; S2 = {{18; 23}; {7; 32}; {28; 33}; {35; 16}; {15; 17}; {12; 3}; {29; 30}; {5; 9}; {34; 1}; {6; 26}; {10; 31}; {19; 22}; {25; 27}; {0; 8}; {4; 13}; {14; 21}; {24; 11}}; S3 = {{1; 26}; {28; 31}; {18; 6}; {8; 30}; {14; 24}; {9; 23}; {15; 0}; {34; 13}; {27; 7}; {12; 25}; {29; 10}; {5; 20}; {16; 32}; {11; 4}; {35; 22}; {17; 33}; {2; 19}}: ∗ GFS(219 ; 2): S0 = {{5; 23}; {27; 17}; {9; 10}; {31; 37}; {11; 25}; {6; 33}; {34; 12}; {22; 30}; {14; 1}; {28; 2}; {15; 13}; {16; 7}; {35; 20}; {8; 3}; {21; 36}; {32; 4}; {29; 18}; {24; 26}}; S1 = {{9; 25}; {33; 36}; {13; 4}; {35; 23}; {37; 34}; {2; 22}; {3; 28}; {18; 19}; {29; 21}; {5; 12}; {0; 17}; {31; 27}; {7; 24}; {15; 8}; {32; 26}; {30; 16}; {14; 10}; {11; 6}}: ∗ GFS(223 ; 2): S0 = {{36; 6}; {14; 29}; {34; 38}; {22; 7}; {3; 41}; {30; 2}; {45; 28}; {4; 9}; {32; 35}; {18; 13}; {5; 8}; {31; 17}; {19; 12}; {16; 43}; {39; 21}; {33; 27}; {20; 26}; {1; 37}; {40; 15}; {10; 11}; {25; 42}; {24; 44}};
394
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
S1 = {{30; 8}; {5; 14}; {38; 3}; {26; 12}; {2; 4}; {13; 40}; {10; 0}; {43; 23}; {42; 34}; {32; 44}; {22; 35}; {9; 11}; {33; 29}; {18; 27}; {19; 31}; {41; 17}; {25; 36}; {37; 7}; {15; 16}; {6; 39}; {21; 28}; {45; 20}}: ∗ GFS(226 ; 4): S0 = {{50; 29}; {39; 44}; {51; 34}; {23; 41}; {40; 11}; {48; 24}; {19; 35}; {47; 25}; {2; 20}; {15; 22}; {21; 38}; {13; 32}; {10; 46}; {28; 42}; {18; 8}; {14; 27}; {49; 3}; {37; 1}; {45; 17}; {9; 6}; {43; 31}; {30; 12}; {5; 7}; {16; 33}; {36; 4}}; S1 = {{28; 6}; {2; 25}; {48; 38}; {20; 16}; {50; 26}; {46; 23}; {18; 43}; {14; 11}; {51; 3}; {39; 9}; {29; 15}; {21; 33}; {13; 22}; {19; 12}; {31; 10}; {24; 8}; {0; 45}; {5; 37}; {32; 47}; {34; 30}; {41; 4}; {42; 44}; {49; 36}; {17; 35}; {40; 7}}; S2 = {{24; 27}; {19; 38}; {49; 39}; {13; 14}; {41; 0}; {46; 6}; {11; 36}; {26; 18}; {3; 45}; {51; 8}; {1; 5}; {12; 23}; {34; 43}; {32; 9}; {17; 42}; {25; 20}; {47; 33}; {22; 7}; {44; 31}; {16; 4}; {15; 35}; {48; 21}; {50; 30}; {10; 29}; {37; 40}}; S3 = {{32; 1}; {2; 17}; {37; 28}; {25; 30}; {19; 40}; {47; 48}; {10; 21}; {46; 51}; {14; 44}; {43; 45}; {42; 31}; {27; 33}; {49; 5}; {50; 4}; {20; 6}; {13; 26}; {36; 38}; {34; 41}; {35; 0}; {11; 39}; {9; 8}; {24; 16}; {12; 18}; {15; 7}; {22; 23}}: ∗ GFS(235 ; 2): S0 = {{9; 63}; {11; 43}; {36; 52}; {67; 10}; {3; 69}; {49; 21}; {37; 39}; {13; 61}; {20; 44}{55; 2}; {66; 7}; {6; 38}; {4; 41}; {47; 48}; {1; 53}; {64; 8}; {19; 46}; {62; 33}; {58; 56}; {23; 29}; {51; 31}; {25; 16}; {57; 45}; {32; 50}; {24; 30}; {60; 65}; {12; 54}; {59; 26}; {17; 40}; {18; 28}; {34; 27}; {42; 22}; {68; 15}; {14; 5}}; S1 = {{53; 8}; {65; 46}; {68; 19}; {43; 69}; {47; 33}; {60; 30}; {42; 64}; {29; 4}; {7; 37}; {40; 41}; {50; 14}; {18; 67}; {52; 56}; {31; 39}; {59; 20}; {21; 6}; {62; 15}; {48; 9}; {22; 51}; {35; 38}; {24; 12}; {0; 55}; {25; 44}; {16; 5}; {10; 17}; {57; 11}; {2; 28}; {58; 61}; {32; 45}; {49; 54}; {63; 27}; {26; 34}; {3; 13}; {23; 66}}: ∗ GS(238 ; 4): S0 = {{67; 10}; {75; 46}; {29; 64}; {5; 15}; {50; 55}; {57; 9}; {23; 30}; {18; 51}; {31; 16}; {17; 1}; {27; 19}; {60; 73}; {62; 20}; {8; 59}; {53; 47}; {13; 32}; {33; 35}; {49; 52}; {45; 25}; {70; 34}; {28; 54}; {22; 4}; {12; 24}; {74; 14}; {3; 36}; {65; 40}; {41; 58}; {63; 42}; {37; 68}; {26; 71}; {7; 48}; {72; 2}; {69; 66}; {61; 6}; {56; 11}; {44; 21}; {43; 39}}; S1 = {{40; 11}; {48; 49}; {68; 3}; {9; 58}; {69; 46}; {5; 57}; {25; 55}; {19; 6}; {66; 72}; {8; 56}; {60; 4}; {54; 2}; {16; 24}; {38; 50}; {63; 29}; {13; 14}; {7; 21}; {15; 52}; {36; 65}; {35; 47}; {44; 26}; {71; 17}; {30; 0}; {20; 74}; {31; 61}; {28; 45}; {34; 62}; {41; 10}; {37; 27}; {33; 18}; {23; 51}; {43; 70}; {67; 64}; {75; 22}; {59; 32}; {12; 73}; {53; 42}}; S2 = {{47; 11}; {23; 26}; {34; 42}; {75; 38}; {73; 71}; {50; 46}; {57; 48}; {68; 13}; {20; 64}; {63; 74}; {45; 9}; {8; 70}; {31; 15}; {69; 25}; {35; 10}; {59; 39}; {19; 0}; {41; 4}; {5; 17}; {1; 30}; {56; 58}; {22; 65}; {61; 54}; {53; 60}; {12; 52}; {62; 72}; {24; 28}; {27; 18}; {37; 32}; {3; 16}; {51; 7}; {6; 36}; {67; 49}; {43; 21}; {66; 44}; {14; 33}; {55; 29}};
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
395
S3 = {{61; 70}; {26; 16}; {28; 42}; {74; 18}; {38; 64}; {49; 15}; {54; 17}; {22; 9}; {25; 7}; {62; 27}; {55; 56}; {14; 46}; {63; 39}; {11; 4}; {34; 35}; {23; 44}; {72; 29}; {30; 32}; {6; 57}; {40; 24}; {50; 67}; {21; 48}; {52; 10}; {43; 20}; {51; 66}; {58; 53}; {47; 33}; {36; 31}; {8; 60}; {0; 65}; {71; 12}; {68; 59}; {19; 45}; {37; 2}; {5; 13}; {73; 1}; {69; 75}}: ∗ GFS(247 ; 2): S0 = {{7; 21}; {30; 20}; {74; 70}; {23; 48}; {75; 40}; {26; 58}; {22; 81}; {91; 41}; {83; 55}; {59; 64}; {3; 12}; {90; 63}; {33; 52}; {61; 32}; {49; 78}; {82; 16}; {66; 6}; {44; 2}; {80; 89}; {84; 10}; {15; 67}; {11; 71}; {45; 38}; {85; 13}; {54; 28}; {57; 17}; {53; 86}; {43; 24}; {5; 42}; {27; 76}; {19; 92}; {35; 29}; {88; 31}; {62; 51}; {72; 14}; {36; 50}; {34; 4}; {1; 18}; {65; 68}; {77; 93}; {87; 60}; {69; 37}; {56; 25}; {8; 79}; {9; 73}; {39; 46}}; S1 = {{56; 54}; {43; 64}; {63; 50}; {60; 36}; {77; 52}; {41; 8}; {39; 75}; {2; 51}; {47; 4}; {29; 49}; {65; 61}; {93; 83}; {19; 37}; {88; 73}; {82; 3}; {62; 9}; {46; 84}; {44; 66}; {27; 53}; {76; 33}; {79; 80}; {57; 34}; {13; 21}; {92; 42}; {58; 10}; {87; 32}; {23; 11}; {20; 38}; {90; 35}; {45; 28}; {12; 0}; {55; 7}; {14; 15}; {78; 70}; {25; 81}; {26; 31}; {71; 69}; {18; 72}; {89; 86}; {67; 91}; {5; 68}; {17; 30}; {24; 40}; {85; 74}; {6; 59}; {16; 22}}: References Beth, T., Jungnickel, D., Lenz, H., 1986. Design Theory. Cambridge University Press, London. Bird, I.C.M., Keedwell, A.D., 1994. Design and applications of optical orthogonal codes — a survey. Bull. ICA 11, 21–44. Blake-Wilson, S., Phelps, K., 1999. Constant weight codes and group divisible design. Designs, Codes and Cryptography 16, 11–27. Brickell, E.F., Wei, V.K., 1987. Optical orthogonal codes and cyclic block designs. Congr. Numer. 58, 175–192. Chen, K., Ge, G., Zhu, L., 1999. Generalized Steiner triple systems with group size ÿve. J. Combin. Des., to appear. Chung, F.P.K., Salehi, J.A., Wei, V.K., 1989. Optical orthogonal codes: design, analysis, and applications. IEEE Trans. Inform. Theory 35, 595–604. Colbourn, C.J., Dinitz, J.H. (Eds.), 1996. The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton. Dinitz, J.H., Stinson, D.R., 1992. Room squares and related designs. in: Contemporary Design Theory: A Collection of Surveys. Wiley, New York, pp. 137–204. Etzion, T., 1997. Optimal constant weight codes over Zk and generalized designs. Discrete Math. 169, 55–82. Phelps, K., Yin, C., 1997. Generalized Steiner systems with block size three and group size g ≡ 3 (mod 6). J. Combin. Des. 5, 417–432. Phelps, K., Yin, C., 1999. Generalized Steiner systems with block size three and size four. Ars Combin., to appear. Salehi, J.A., 1989. Emerging optical code-division multiple access communications systems. IEEE Network 3–2, 31–39. Salehi, J.A., Brackett, C.A., 1989. Code-division multiple access techniques in optical ÿber networks: parts 1 and 2. IEEE Trans. Commun. 37, 824–842. Wu, D., Zhu, L., Ge, G., 1999. Generalized Steiner triple systems with group size g = 7; 8. Ars Combin., to appear. Yin, J., 1998. Some combinatorial constructions for optical orthogonal codes. Discrete Math. 185, 201–219.
www.elsevier.com/locate/jspi
Starters and related codes K. Chen, G. Ge, L. Zhu∗ Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China Received 28 March 1998; received in revised form 24 June 1998; accepted 30 October 1998
In Honor of Professor R.G. Stanton
Abstract Starters and their connections with codes are discussed. It is shown that there exists a skew starter in Zv for all v such that gcd(v; 6) = 1, v is either not divisible by 5 or divisible by 25. This can be used to show that there exists an optimal (6v; 4; 1) optical orthogonal code in Z6v for all v such that gcd(v; 6) = 1. By using frame starters with some special properties and their generalization, the spectrum of length n is determined to be n¿8 for maximum constant weight c 2000 Elsevier Science B.V. codes of distance 3 and weight 3 over an alphabet of size 7. All rights reserved. MSC: 05B07; 94B25 Keywords: Skew starter; Optical orthogonal code; Generalized Steiner triple system; Maximum constant weight code; Frame starter
1. Introduction and main results Let G be an additive abelian group of order v. A starter in G is a set of unordered pairs S = {{xi ; yi }: 16i6(v − 1)=2} which satisÿes the following two properties: (1) {xi : 16i6(v − 1)=2} ∪ {yi : 16i6(v − 1)=2} = G\{0}: (2) {±(xi − yi ): 16i6(v − 1)=2} = G\{0}. Clearly, there exists a starter in G only if v is odd. A starter S = {{xi ; yi }: 16i6(v − 1)=2} is said to be skew if it further satisÿes the following property: (3) {±(xi + yi ): 16i6(v − 1)=2} = G\{0}. Skew starters have been useful in constructions of Room squares and other combinatorial designs (see Dinitz and Stinson, 1992). Skew starters in Zv have special
Research supported in part by NSFC Grant 19831050. Corresponding author. E-mail address: [email protected] (L. Zhu) ∗
c 2000 Elsevier Science B.V. All rights reserved. 0378-3758/00/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 9 9 ) 0 0 1 1 9 - 6
380
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
application to constructions of Hamiltonian path balanced tournament designs (see Dinitz and Stinson, 1992). With the following known results, Dinitz and Stinson presented an open problem in Dinitz and Stinson (1992). Lemma 1.1 (Dinitz and Stinson, 1992). There is a skew starter in Zv for all v ¿ 5 if v is not divisible by 2; 3 and 5. There does not exist any skew starter in Zv if v ≡ 0 (mod 3). Open Problem (Dinitz and Stinson, 1992, Open Problem 2). Prove that there exists a skew starter in Zv for all v such that gcd(v; 6) = 1. In this article, we shall give a better answer than Lemma 1.1 to the open problem. Speciÿcally, we shall prove the following. Theorem 1.2. There exists a skew starter in Zv for all v such that gcd(v; 6) = 1; v is not divisible by 5 or v is divisible by 25. The results on skew starters in Zv can be used to discuss the existence of optical orthogonal codes. Let v; k and be positive integers. Following Brickell and Wei (1987), a (v; k; ) optical orthogonal code (OOC), C, is a family of (0; 1)-sequences of length v and weight k satisfying the following two properties: P (1) the auto-correlation property: 06t6v−1 xt xt+i 6, for any x =(x0 ; x1 ; : : : ; xv−1 ) ∈ C and any integer i 6≡ 0 (mod v); P (2) the cross-correlation property: 06t6v−1 xt yt+i 6, for any x 6= y in C and any integer i. Here, all subscripts are reduced modulo v. The study of OOCs is motivated by an application in a code-division multiple access ÿber optical channel which requires binary sequences with good correlation properties. For related details, the interested reader may refer to Brickell and Wei (1987), Chung et al. (1989), Salehi and Brackett (1989) and Salehi (1989). There are some combinatorial constructions for OOCs, especially for = 1, (see, for example, Bird and Keedwell, 1994; Yin, 1998). A (v; k; 1) OOC with |C| = b(v − 1)=k(k − 1)c codewords is said to be optimal. It was shown in Brickell and Wei (1987) that an optimal (v; 3; 1) OOC exists if and only if v 6= 6t + 2 and t ≡ 2 or 3 (mod 4). When k¿4, the existence problem for an optimal (v; k; 1) OOC remains unsolved. We shall use skew starters in Zv to construct optimal (6v; 4; 1) optical orthogonal code in Z6v and show the following. Theorem 1.3. There exists an optimal (6v; 4; 1) optical orthogonal code in Z6v for all v such that gcd(v; 6) = 1. In Section 3, we shall discuss some frame starters with special properties. This discussion leads to a class of maximum constant weight codes. A (g + 1)-ary constant weight code (n; w; d) is a code C ⊆(Zg+1 )n of length n and minimum distance d, such that every c ∈ C has Hamming weight w. To construct a constant weight code (n; w; d) with w = 3 a group divisible design (GDD) is used. A K − GDD(gn ) is an ordered
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
381
triple (P; G; B) where P is a set of gn elements, G is a collection of g-subsets called groups which partition P, and B is a set of some subsets called blocks of P, such that each block intersects each group in at most one element and that each pair of elements from distinct groups occurs together in exactly one block in B, where |B| ∈ K for any B ∈ B. In a 3 − GDD(gn ), let P = (Zg+1 \ {0}) × (Zn+1 \ {0}) with n groups Gi ∈ G; Gi = (Zg+1 \{0}) × {i}; 16i6n and blocks {(a; i); (b; j); (c; k)} ∈ B. One can construct a constant weight code (n; 3; d) as stated in Phelps and Yin (1997). From each block we form a codeword of length n by putting an a; b and c in positions i, j and k; respectively, and zeros elsewhere. This gives a constant weight code over Zg+1 with minimum distance 2 or 3. If the minimum distance is 3, then the code is a (g + 1)-ary maximum constant weight code (MCWC) (n; 3; 3) and the 3 − GDD(gn ) is called generalized Steiner triple system, denoted by GS(2; 3; n; g). The following is known. Lemma 1.4 (Etzion, 1997; Phelps and Yin, 1997). The necessary conditions for the existence of GS(2; 3; n; g) (or a (g + 1)-ary MCWC(n; 3; 3)) is that: (1) if g ≡ 0 (mod 6); then n¿g + 2; (2) if g ≡ 3 (mod 6); the n ≡ 1 (mod 2) and n¿g + 2; (3) if g ≡ 2 or 4 (mod 6); then n ≡ 0 or 1 (mod 3) and n¿g + 2; (4) if g ≡ 1 or 5 (mod 6); then n ≡ 1 or 3 (mod 6) and n¿g + 2. Lemma 1.5 (Etzion, 1997; Phelps and Yin, 1997; Phelps and Yin, to appear; Chen et al., to appear). There exists a (g + 1)-ary MCWC(n; 3; 3) for g = 2; 3; 4; 5 and 9 if and only if the necessary conditions in Lemma 1:4 are satisÿed with one exception of (g; n) = (2; 6). Recently, Blake-Wilson and Phelps (1999) proved that the necessary conditions listed in Lemma 1.4 are asymptotically sucient for any g. In this article, we shall show the following. Theorem 1.6. There exists a (g + 1)-ary MCWC(n; 3; 3) for g = 6 if and only if n¿8.
2. Skew starter and OOC In this section, we shall give the proof of Theorems 1.2 and 1.3. Lemma 2.1. There exists a skew starter in Z25 . Proof. We take S as follows: S = {{1; 2}; {3; 5}; {4; 7}; {6; 20}; {8; 13}; {9; 22}; {10; 17}; {11; 21}; {12; 18}; {14; 23}; {15; 19}; {16; 24}}: It is readily checked that S is a skew starter in Z25 .
382
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
Lemma 2.2. There exists a skew starter in Z125 . Proof. With the aid of a computer, we have found the following skew starter S in Z125 : S = {{1; 43}; {2; 49}; {3; 102}; {4; 116}; {5; 68}; {6; 117}; {7; 95}; {8; 60}; {9; 98}; {10; 11}; {12; 35}; {13; 67}; {14; 20}; {15; 70}; {16; 77}; {17; 122}; {18; 53}; {19; 115}; {21; 66}; {22; 61}; {23; 91}; {24; 45}; {25; 107}; {26; 103}; {27; 106}; {28; 38}; {29; 79}; {30; 58}; {31; 123}; {32; 83}; {33; 120}; {34; 92}; {36; 105}; {37; 52}; {39; 71}; {40; 124}; {41; 81}; {42; 118}; {44; 56}; {46; 55}; {47; 51}; {48; 108}; {50; 69}; {54; 84}; {57; 101}; {59; 112}; {62; 121}; {63; 74}; {64; 80}; {65; 82}; {72; 94}; {73; 78}; {75; 100}; {76; 110}; {85; 93}; {86; 104}; {87; 114}; {88; 119}; {89; 113}; {90; 97}; {96; 99}; {109; 111}}: Lemma 2.3. If there exist skew starters in Zn and Zm ; respectively; then there exists a skew starter in Znm . Proof. Let S1 = {{xi ; yi }: 16i6(n − 1)=2} and S2 = {{at ; bt }: 16t6(m − 1)=2} be skew starters in Zn and Zm , respectively. Let F = {{xi + jn; yi + 2jn}: 16i6(n − 1)=2; 06j6m − 1}; and S = F ∪ {{at n; bt n}: 16t6(m − 1)=2}: Notice that gcd(m; 6) = 1, it is easy to check that S is a skew starter in Znm . We can now prove Theorem 1.2. Proof of Theorem 1.2. If v is not divisible by 5, then the result comes from Lemma 1.1. If v is divisible by 25, then we can write v as v = 5k t, where k¿2 and gcd(t; 30) = 1. By Lemmas 2.1–2.3 we know that there exists a skew starter in Z5k . The existence of a skew starter in Zt is guaranteed by Lemma 1.1. So, again apply Lemma 2.3, we get the result. To answer the open problem mentioned in Section 1 completely, by Theorem 1.2, one needs only to consider the case v = 5t, where gcd(t; 30) = 1. In this case, the form v = 5p is fundamental where p ¿ 5 is an odd prime. For v ¡ 100 of this form, we have done a computer search and obtained the following. Lemma 2.4. There exists a skew starter in Zv for any v ∈ {35; 55; 65; 85; 95}. Proof. With the aid of a computer, we have found the following skew starters: v= 35; {1; 2}; {3; 5}; {4; 7}; {6; 10}; {8; 15}; {9; 21}; {11; 25}; {12; 29}; {13; 24}; {14; 30}; {16; 26}; {17; 22}; {18; 31}; {19; 34}; {20; 28}; {23; 32}; {27; 33}:
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
383
v= 55; {1; 3}; {2; 37}; {4; 40}; {5; 44}; {6; 46}; {7; 34}; {8; 9}; {10; 15}; {11; 18}; {12; 20}; {13; 43}; {14; 23}; {16; 26}; {17; 28}; {19; 45}; {21; 39}; {22; 35}; {24; 38}; {25; 49}; {27; 50}; {29; 41}; {30; 33}; {31; 52}; {32; 54}; {36; 53}; {42; 48}; {47; 51}: v= 65; {1; 42}; {2; 40}; {3; 23}; {4; 5}; {6; 10}; {7; 62}; {8; 44}; {9; 32}; {11; 51}; {12; 58}; {13; 45}; {14; 16}; {15; 18}; {17; 38}; {19; 31}; {20; 25}; {21; 52}; {22; 61}; {24; 35}; {26; 41}; {27; 49}; {28; 56}; {29; 64}; {30; 36}; {33; 46}; {34; 48}; {37; 53}; {39; 57}; {43; 60}; {47; 54}; {50; 59}; {55; 63}: v= 85; {1; 24}; {2; 72}; {3; 62}; {4; 40}; {5; 78}; {6; 73}; {7; 47}; {8; 42}; {9; 39}; {10; 57}; {11; 71}; {12; 68}; {13; 63}; {14; 15}; {16; 84}; {17; 38}; {18; 50}; {19; 43}; {20; 26}; {21; 52}; {22; 31}; {23; 66}; {25; 27}; {28; 36}; {29; 49}; {30; 69}; {32; 80}; {33; 74}; {34; 41}; {35; 51}; {37; 64}; {44; 77}; {45; 59}; {46; 65}; {48; 61}; {53; 75}; {54; 82}; {55; 58}; {56; 67}; {60; 70}; {76; 81}; {79; 83}: v= 95; {1; 37}; {2; 77}; {3; 61}; {4; 82}; {5; 72}; {6; 67}; {7; 34}; {8; 43}; {9; 17}; {10; 60}; {11; 85}; {12; 50}; {13; 26}; {14; 44}; {15; 84}; {16; 71}; {18; 94}; {19; 63}; {20; 69}; {21; 54}; {22; 24}; {23; 79}; {25; 89}; {27; 33}; {28; 46}; {29; 32}; {30; 62}; {31; 74}; {35; 49}; {36; 45}; {38; 42}; {39; 68}; {40; 87}; {41; 52}; {47; 88}; {48; 90}; {51; 73}; {53; 78}; {55; 70}; {56; 66}; {57; 80}; {58; 65}; {59; 83}; {64; 76}; {75; 91}; {81; 86}; {92; 93}: At present, we are not able to determine the existence of a skew starter of this case completely. However, we may construct some optimal (6v; 4; 1) OOC by using the known results obtained above. We need the concept of cyclic packing (see Yin, 1998). Let B = {B1 ; B2 ; : : : ; Bt }, where Bi = {bi1 ; bi2 ; : : : ; bik }, bij ∈ Zv , 16i6t and 16j6k. Let D = {bij − bis : 16i6t; j 6= s; 16j; s6k}. The pair (Zv ; B) is said to be a cyclic packing CP(v; k; 1) if the cardinality of D is k(k − 1)t and 0 6∈ D. Further, it is called optimal when t = b(v − 1)=k(k − 1)c. The following result was presented in Yin (1998). Lemma 2.5 (Yin, 1998). The existence of an optimal (v; k; 1) OOC is equivalent to the existence of an optimal CP(v; k; 1). Lemma 2.6. If there exists a skew starter in Zv such that gcd(v; 6) = 1; then there exists an optimal (6v; 4; 1) OOC. Proof. Let S = {{xi ; yi }: 16i6(v − 1)=2} be a skew starter in Zv , where v = 2t + 1. In Zv × Z6 , which is isomorphic to Z6v since gcd(v; 6) = 1, let Bi = {(xi ; 0); (yi ; 0); (xi + yi ; 1); (0; 4)}, 16i6t. We show that the 12t di erences provided by Bi are nonzero and di erent from each other. These di erences can be divided into 6 classes according to the second coordinate: (1) ±(xi − yi ; 0), 16i6t, (2) (yi ; 1), (xi ; 1), 16i6t,
384
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
(3) (xi ; 2), (yi ; 2), 16i6t, (4) (xi + yi ; 3), (−xi − yi ; 3), 16i6t, (5) (−xi ; 4), (−yi ; 4), 16i6t, (6) (−yi ; 5), (−xi ; 5), 16i6t. By the deÿnition of S, it is easily seen that the 2t di erences in each of the classes (1) – (6) are nonzero and di erent from each other as expected. It follows that Bi (16i6t) form a CP(6v; 4; 1) in Zv ×Z6 . Since b(6v −1)=4(4−1)c =b(12t +5)=12)c =t, so the cyclic packing is optimal. By Lemma 2.5, there exists an optimal (6v; 4; 1) OOC. Lemma 2.7. There exists an optimal CP(6v; 4; 1) for all v = 5h; h ¿ 1 and gcd(h; 30) = 1. Proof. Since gcd(h; 30) = 1, by Lemma 1:2, there exists a skew starter in Zh : S = {{xi ; yi }: 16i6(h − 1)=2}: Let F = {{x + jh; y + 2jh}: {x; y} ∈ S; 06j64}; in Zv and let B = {{( ; 0); ( ; 0); ( + ; 1); (0; 4)}: { ; } ∈ F}; A = {{(0; 0); (h; 1); (3h; 3); (0; 4)}; {(0; 0); (0; 5); (3h; 1); (4h; 1)}}; in Zv × Z6 . Then B ∪ A is an optimal CP(6v; 4; 1) in cyclic group Zv × Z6 . In fact, the 4-subsets in B provide 12 × 5 × (h − 1)=2 di erences which form the set (Z5h\{0; h; 2h; 3h; 4h})×Z6 . Meanwhile, the 4-subsets in A provide 24 di erences which form the set ({0; h; 2h; 3h; 4h} × Z6 )\{(0; 0); (0; 3); ±(h; 2); ±(2h; 0)}. So the result is obtained. We can now prove Theorem 1.3. Proof of Theorem 1.3. If v = 5t; t ¿ 1 and gcd(t; 5) = 1, then the result comes from Lemmas 2.7 and 2.5. Otherwise, v is not divisible by 5 or is divisible by 25, there exists a skew starter in Zv by Theorem 1.2. So we get the result by Lemma 2.6. 3. Frame starters and MCWC An important generalization of a starter is the frame starter. In this section we shall use frame starters with some additional properties ∗ FS(2n ) and their variations ∗ GFS(2n ) to construct GS(2; 3; n; 6) for small values of n. We shall also apply PBD construction to prove Theorem 1.6. Let G be an additive abelian group of order nh and H be a subgroup of order h, where nh − h is even. A frame starter in G\H of type hn (simply, FS(hn )) is a set of
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
385
unordered pairs S = {{xi ; yi }: 16i6(nh − h)=2} such that the following two properties are satisÿed: (1) {xi : 16i6(nh − h)=2} ∪ {yi : 16i6(nh − h)=2} = G\H . (2) {±(xi − yi ): 16i6(nh − h)=2} = G\H . Thus, a starter is the special case of a frame starter when H = {0}. From an FS(hn ) one may easily get a 3−GDD((3h)n ) (P; G; B) as follows: P = G × Z3 ; G = {G1 ; G2 ; : : : ; Gn }; Gj = Hj × Z3 ; Hj is the jth coset of H in G, and Bi = {(xi ; 0); (yi ; 0); S (0; 1)}; 16i6(nh − h)=2, B = 16i6(nh−h)=2 {Bi + (a; b): a ∈ G; b ∈ Z3 }. It is clear that the corresponding constant weight code contains two codewords having distance 2 i the 3−GDD contains two intersecting blocks which intersect only three groups of the 3−GDD. By the above construction this can happen only if the blocks {(0; 0); (yi − xi ; 0); (−xi ; 1)}; {(xi − yi ; 0); (0; 0); (−yi ; 1)}; {(xi ; 2); (yi ; 2); (0; 0)}; 16i6(nh − h)=2; contain such a pair of blocks. This means that the pairs {hyi − xi in ; h−xi in }; {hxi − yi in ; h−yi in }; {hxi in ; hyi in } for 16i6(nh − h)=2 cannot be all distinct, where hxin = j when x ∈ Gj . Speciÿcally, hxin ≡ x (mod n) when x ∈ G = Zhn , and h(x; t)in = x when (x; t) ∈ G = Zn × Zh . If these pairs are all distinct, then we obtain a GS(2; 3; n; 3h). Denote such an FS(hn ) by ∗ FS(hn ). Thus, we have proved the following: Lemma 3.1. If there exists a
∗
FS(hn ); then there exists a GS(2; 3; n; 3h).
In the remainder of this article we shall focus on the existence of a GS(2; 3; n; 6). From Lemma 3.1, we need to discuss the existence of ∗ FS(2n ). Lemma 3.2. There exists a
∗
FS(28 ) in Z16 \{0; 8}.
Proof. Let S = {{1; 2}; {3; 5}; {4; 10}; {6; 13}; {7; 11}; {9; 14}; {12; 15}}: It is readily checked that S is a ∗ FS(28 ) in Z16 \{0; 8}. Let [a; b] denote the set of integers in the interval a6n6b. We shall discuss the method of PBD-closure which reduces the existence of a GS(2; 3; n; 6) to the set E = [8; 79]\{57; 64; 65; 72; 73, 74}. We are quite successful in ÿnding a ∗ FS(2n ) for most values n ∈ E1 = {v ∈ E: v ≡ 0; 1(mod 4)}, excepting for n = 9. To ÿnd a ∗ FS(2n ) in Z2n \{0; n} for the remaining values n of E1 , we can take a suitable multiplier m ∈ Z2n \{0; n} and a suitable integer t, and then try to ÿnd two St−1 sets of pairs, P0 and R, so that S = P ∪ R forms a ∗ FS(2n ), where P = i=0 Pi and Pi = mi P0 ; 16i6t − 1. We call such a set P0 partial set and R remainder. Lemma 3.3. There exists a
∗
FS(2n ) in Z2n \{0; n} for all n ∈ {12; 16; 20}.
386
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
Proof. With the aid of a computer, we have found a ∗ FS(2n ) in Z2n \ {0; n} for all n ∈ {12; 16; 20}. As mentioned above, for a ∗ FS(2n ), we only list the corresponding multiplier m, integer t, partial set P0 and remainder R as follows. ∗ FS(212 ); m = 3; t = 2; P0 = {{1; 2}}; R = {{4; 9}; {5; 11}; {7; 14}; {8; 19}; {10; 20}; {13; 22}; {15; 23}; {16; 18}; {17; 21}}: ∗ FS(216 ); m = 3; t = 4; P0 = {{1; 2}}; R = {{8; 20}; {10; 23}; {12; 30}; {14; 21}; {17; 28}; {7; 24}; {5; 31}; {13; 15}; {4; 26}; {11; 19}; {25; 29}}: ∗ FS(220 ); m = 3; t = 4; P0 = {{1; 2}; {4; 11}}; R = {{21; 25}; {5; 30}; {7; 29}; {8; 13}; {26; 34}; {10; 16}; {24; 38}; {22; 32}; {15; 31}; {23; 35}; {37; 39}}: Now, we discuss the method of PBD-closure. A pairwise balanced design of order v ((v; K)-PBD) is an ordered pair (X; B), where X is a set (of points) of size v and B is a collection of subsets of X (called blocks) with the property that every pair of elements of X occurs together in exactly one block B ∈ B, |B| ∈ K. Let B(K) denote the PBD-closure of K, i.e., the set of positive integers n for which there exists a PBD(n; K). K is PBD-closed if B(K) = K. It is well known (see Beth et al., 1986) that if K is a closed set, then there exists a ÿnite subset J ⊆ K such that K = B(J ). This set J is called a generating set for the PBD-closed set K. Let T = {n: there exists a GS(2; 3; n; 6)}. By the deÿnition of a GS(2; 3; n; g) and the well-known Wilson’s fundamental construction for GDDs (see Colbourn and Dinitz, 1996, III Theorem 2.5), it is easy to see that T is PBD-closed. Let K8 be the set of positive integers n¿8. By the necessary conditions stated in Lemma 1.4, we have T ⊆ K8 . To prove Theorem 1.6, it suces to prove that K8 ⊆ T . To do this, we need only to show that E is a generating set for K8 and that E ⊆ T . Lemma 3.4 (Colbourn and Dinitz, 1996). K8 =B(E); where E =[8; 79]\{57; 64; 65; 72; 73; 74}. Let E2 = E \ E1 . That is, E2 = {v ∈ E: v ≡ 2; 3 (mod 4)}. It is not dicult to see that there is no FS(2n ), consequently, no ∗ FS(2n ) in Z2n \{0; n} for n ≡ 2 or 3 (mod 4) (see Dinitz and Stinson, 1992, Theorem 2:14). So, the method given in Lemma 3.1 is failed to give a GS(2; 3; n; 6) for any n ∈ E2 . We need a generalized concept of FS(hn ) and ∗ FS(hn ). Let H be a subgroup of order h of Znh , where nh − h is even. Let w be a positive integer such that w|nh. A generalized frame starter in Znh \ H , with respect to w, Sw−1 of type (hn ) (simply, GFS(hn ; w)), is a set of unordered pairs S = j=0 Sj , where Sj = {{xij ; yij }: 16i6(nh − h)=2}; 06j6w − 1, such that the following two properties are satisÿed: (1) {xij − j: 16i6(nh − h)=2} ∪ {yij − j: 16i6(nh − h)=2} = Znh \H; 06j6w − 1, Sw−1 (2) j=0 ({(xij −yij ; hxij iw ): 16i6(nh−h)=2}∪{(yij −xij ; hyij iw ): 16i6(nh−h)=2})= (Znh \H ) × Zw . Let aij =hxij −jin ; bij =hyij −jin , and cij =hyij −xij in for 16i6(nh−h)=2, 06j6w−1. S is called a ∗ GFS(hn ; w) if it further satisÿes the following property that in Zn ×Zn ×Zw ,
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
387
all of the following 3w(nh − h) ordered triples (aij ; bij ; hjiw ); (bij ; aij ; hjiw ); (cij ; n − aij ; hxij iw ); (n − aij ; cij ; hxij iw ); (n − cij ; n − bij ; hyij iw ); (n − bij ; n − cij ; hyij iw ) for 16i6(nh − h)=2; 06j6w − 1, are distinct. Obviously, a GFS(hn ; 1) (or ∗ GFS(hn ; 1)) is just an FS(hn ) (or ∗ FS(hn )). For any w such that w|nh, an FS(hn ) (or ∗ FS(hn )) is also a GFS(hn ; w) (or ∗ GFS(hn ; w)). We have a construction for a GS(2; 3; n; 3h) which is similar to that of Lemma 3.1. Lemma 3.5. If there exists a ∗ GFS(hn ; w) for some positive integer w such that w|nh; then there exists a GS(2; 3; n; 3h). Sw−1 Proof. Let S = j=0 Sj be a ∗ GFS(hn ; w) in Znh \ H , where Sj = {{xij ; yij }: 16i6 (nh − h)=2}; {06j6w − 1}. Let Bij = {(xij ; 0); (yij ; 0); (j; 1)}; 16i6(nh − h)=2; { 06j6w − 1}; and B=
(nh−h)=2 w−1 [ [ i=1
{Bij + (aw; b): 16a6nh=w; b ∈ Z3 }:
j=0
We shall show that (Znh × Z3 ; G; B) is a 3−GDD((3h)n ), where G = {G0 ; G1 ; : : : ; Gn−1 }, and Gi = (H + i) × Z3 (06i6n − 1) are groups. In fact, |B| =3h2 n(n − 1)=2. So, it suces to show that for any two groups Gs and Gt , s 6= t, and for any ( ; ) ∈ Gs ; ( ; ) ∈ Gt , there exists a block A ∈ B such that ( ; ) and ( ; ) lie in A together. We shall prove this in two cases. (i) If = , then by the deÿnition of a GFS(hn ), we know that there must exist a pair {x; y} ∈ Sj for some j; 06j6w − 1, such that (x − y; hxiw ) = ( − ; h iw ): Since hxiw = h iw , we have − x = − y = aw for some a ∈ Znh . This forces (y − x; hyiw ) = ( − ; h iw ): Take A = {(x; 0); (y; 0); (j; 1)} + (aw; ). Obviously, A ∈ B and ( ; ) and ( ; ) lie in A. (ii) If 6= , without loss of generality, we may assume that − = 1 in Z3 . Let j = h iw ; then − j = aw for some a ∈ Znh . Since ( ; ) ∈ Gs ; ( ; ) ∈ Gt ; s 6= t, we have − ∈ Znh \H . So, − j − aw ∈ Znh \H . By the deÿnition of a GFS(hn ), there exists a pair {x; y} ∈ Sj such that x − j = − j − aw, or y − j = − j − aw, i.e. x (or y)= − aw in Znh . Take A = {(x; 0); (y; 0); (j; 1)} + (aw; ) in B. Clearly, ( ; ) and ( ; ) lie in A. Next we show that the 3−GDD is a GS(2; 3; n; 3h). Otherwise, the 3−GDD contains two blocks which have a common point and intersect three groups only. Suppose
388
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
that the two blocks are {(x; i); (y; j); (z; k)} and {(x; i); (y; j0 ); (z; k 0 )}, where y ≡ y (mod n); z ≡ z (mod n). Without loss of generality, we have in the ∗ GFS(hn ; w) two pairs {xpq ; ypq } ∈ Sq and {xst ; yst } ∈ St such that x = xpq + aw; y = xpq + aw; z = q + aw; {x; y; z} = {xst + cw; yst + cw; t + cw} for some a; c ∈ Znh . We consider only one special case here, and the other possibilities are similarly proved. Suppose x = xst + cw;
y = yst + cw;
z = t + cw:
Thus, we have ypq − xpq = y − x ≡ y − x = yst − xst (mod n); q − xpq = z − x ≡ z − x = t − xst (mod n); xpq ≡ xst (mod w): This means that the two ordered triples (hypq − xpq in ; hq − xpq in ; hxpq iw ) and (hyst − xst in ; ht − xst in ; hxst iw ) are the same. This contradicts the deÿnition of a ∗ GFS(hn ). ∗
From this, we know that to construct a GS(2; 3; n; 6) for n ∈ E2 , it suces to ÿnd a GFS(2n ; w) for some positive integer w such that w|2n. We have the following.
Lemma 3.6. There exists a
∗
GFS (29 ; 3) in Z18 \{0; 9}.
Proof. With the aid of a computer, a ∗ GFS(29 ; 3) in Z18\{0; 9} has been found, which is listed as follows: S0 = {{16; 12}; {1; 3}; {15; 2}; {7; 13}; {14; 6}; {17; 5}; {10; 11}; {8; 4}}; S1 = {{11; 4}; {5; 13}; {2; 3}; {16; 6}; {14; 17}; {15; 9}; {7; 0}; {8; 12}}; S2 = {{17; 6}; {15; 16}; {13; 0}; {9; 12}; {4; 7}; {5; 3}; {1; 14}; {10; 8}}: Lemma 3.7. For any n ∈ F = {10; 11; 14; 15; 18; 19; 23; 26; 35; 38; 47}; there exists a ∗ GFS (2n ; w) in Z2n \{0; n} for some positive integer w such that w|2n. Proof. For each n ∈ F, with the aid of a computer, we have found a ∗ GFS(2n ; w) for some positive integer w such that w|2n. Here we only list the ∗ GFS(2n ; w)s for n ∈ {10; 11; 14; 15}. For other values in F, we list the corresponding ∗ GFS(2n ; w)s in the appendix. ∗ GFS(210 ; 4): S0 = {{12; 19}; {4; 7}; {9; 6}; {13; 2}; {15; 16}; {18; 11}; {3; 14}; {1; 17}; {5; 8}}; S1 = {{16; 10}; {5; 12}; {3; 9}; {13; 18}; {0; 4}; {2; 14}; {7; 19}; {6; 8}; {17; 15}}; S2 = {{7; 1}; {3; 6}; {4; 13}; {0; 11}; {17; 9}; {16; 8}; {18; 5}; {14; 10}; {15; 19}}; S3 = {{9; 4}; {8; 10}; {16; 11}; {2; 7}; {17; 19}; {5; 6}; {18; 12}; {14; 15}; {0; 1}}: ∗ GFS(211 ; 2): S0 = {{7; 21}; {15; 14}; {17; 12}; {8; 5}; {20; 10}; {2; 18}; {16; 1}; {19; 6}; {3; 9}; {13; 4}};
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
389
S1 = {{2; 17}; {19; 16}; {10; 18}; {14; 9}; {5; 6}; {11; 21}; {13; 15}; {0; 20}; {3; 7}; {4; 8}}: ∗ GFS(214 ; 4): S0 = {{4; 5}; {1; 6}; {23; 16}; {21; 18}; {19; 10}; {22; 27}; {26; 13}; {15; 2}; {11; 7}; {24; 8}; {12; 20}; {25; 17}; {9; 3}}; S1 = {{23; 5}; {21; 17}; {19; 16}; {8; 14}; {22; 11}; {18; 25}; {27; 6}; {2; 12}; {9; 10}; {0; 13}; {24; 26}; {3; 4}; {7; 20}}: S2 = {{15; 20}; {22; 5}; {26; 10}; {17; 23}; {1; 24}; {19; 7}; {0; 11}; {14; 18}; {6; 3}; {8; 12}; {25; 27}; {21; 9}; {4; 13}}; S3 = {{21; 2}; {6; 26}; {14; 4}; {5; 8}; {1; 27}; {16; 9}; {24; 13}; {20; 11}; {10; 12}; {18; 19}; {25; 7}; {15; 23}; {22; 0}}: ∗ GFS(215 ; 2): S0 = {{25; 18}; {7; 29}; {13; 8}; {14; 2}; {21; 17}; {27; 10}; {28; 22}; {19; 9}; {11; 20}; {23; 26}; {1; 12}; {6; 4}; {5; 3}; {16; 24}}; S1 = {{9; 28}; {13; 27}; {15; 3}; {23; 10}; {0; 14}; {7; 4}; {24; 19}; {21; 20}; {6; 29}; {2; 12}; {11; 5}; {17; 8}; {25; 26}; {18; 22}}: For the remaining values n in E, we can also ÿnd a ∗ FS(2n ) when n ∈ E1 or a GFS(2n ; w) when n ∈ E2 , consequently getting a GS(2; 3; n; 6). We had done so in the ÿrst version of this paper. To save space we now deal with these values in other ways, which are described below. It is natural to consider the possibility of using Wilson’s fundamental construction for GDDs to construct GS(2; 3; n; g)s. This construction is also called weighting construction, see (Beth et al., 1986, Theorem IX. 3.2) or (Colbourn and Dinitz, 1996, Theorem III. 2.5). However, the resulting 3−GDD may not have the property that any two intersecting blocks intersect at most two common groups even though the ingredient 3−GDDs do. The reason is that the master GDD may have two intersecting blocks which intersect more than two common groups. In this case, the two ingredient 3−GDDs may contain a block from each one violating the required property. This analysis suggests that we may enforce conditions on the master GDD to avoid such things to happen. This leads to the following deÿnition. Deÿnition 3.1. A K−GDD is said to have “star” property and denoted by K−∗ GDD if any two intersecting blocks intersect at most two common groups. With this deÿnition a GS(2; 3; n; g) is just the same as a 3−∗ GDD(gn ). Using a K−∗ GDD as a master GDD, the well-known Wilson’s fundamental construction can be used to construct GS(2; 3; n; g)s, which we state below. Lemma 3.8 (Weighting). Let (V; G; B) be a K −∗ GDD (the master GDD) with groups G1 ; G2 ; : : : ; Gt . Suppose there exists a function w: V → Z + ∪ {0} (a weighting function) which has the property that for each block B ={x1 ; x2 ; : : : ; xk } ∈ B there exists a 3−∗ GDD of group type (w(x1 ); w(x2 ); : : : ; w(xk )) (such a GDD is an “ingredient”
390
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
P
GDD). Then there exists a 3 −∗ GDD of group type ( P x∈Gt w(x)).
x∈G1
w(x);
P x∈G2
w(x); : : : ;
Proof. The original Wilson’s fundamental construction guarantees that the resultant design is a 3 − GDD. To see the star property we take two blocks from the resultant 3 − GDDs. If they belong to di erent ingredient 3 − GDDs, they must have the star property since the master K−∗ GDD does. Otherwise, they belong to the same ingredient 3 −∗ GDD. So, they must also have the star property. As a corollary the following lemma is useful. Lemma 3.9. If there exists a K −∗ GDD(gn ); and there exists a GS(2; 3; k; h) for any k ∈ K; then there exists a GS(2; 3; n; gh). Proof. In Lemma 3.8, take the K −∗ GDD(gn ) as the master GDD and the GS(2; 3; k; h)s as ingredient 3 −∗ GDDs, the weighting function is w(x) = h for any element x in the master GDD. Lemma 3.10. There exists a 4 −∗ GDD(3n ) for n ∈ {13; 17; 21}. Proof. With the aid of a computer, we have found a set of base blocks A for such a 4 −∗ GDD(3n ) in Z3n with the groups Gi = {i; n + i; 2n + i}, 06i6n − 1, which is listed as follows. n = 13; A = {{0; 1; 6; 31}; {0; 2; 12; 23}; {0; 3; 7; 22}}: n = 17; A = {{0; 1; 3; 8}; {0; 4; 18; 30}; {0; 6; 19; 29}; {0; 9; 20; 36}}: n = 21; A = {{0; 1; 3; 7}; {0; 5; 13; 35}; {0; 9; 20; 38}; {0; 10; 24; 47}; {0; 12; 31; 48}}. By Lemma 1.5 we know that there exists a GS(2; 3; 4; 2). So, by Lemmas 3.10 and 3.9 we have the following. Lemma 3.11. There exists a GS(2; 3; n; 6) for n ∈ {13; 17; 21}. So far, we have proven the following. Lemma 3.12. There exists a GS(2; 3; n; 6) for n ∈ [8; 21] ∪ {23; 26; 35; 38; 47}. Etzion in 1997 established some product constructions for GS(2; 3; n; g)s, some of which can be stated as follows. Lemma 3.13 (Etzion, 1997, Constructions C and D). Let (V; G; B) be a 3 − GDD (gm ). Suppose there exists a GS(2; 3; n; g). Then there exists a GS(2; 3; mn; g) or a GS(2; 3; m(n − 1) + 1; g) if B can be partitioned into t sets S0 ; S1 ; : : : ; St−1 ; such that the minimum distance in Sr , 06r6t − 1; is 3; and t6n or t6n − 1; respectively.
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
391
Lemma 3.14. There exists a 3 − GDD(63 ) with the property that all blocks of the design can be partitioned into t sets S0 ; S1 ; : : : ; St−1 such that t67 and the minimum distance in Sr ; 06r6t − 1, is 3. Proof. Let V = Z18 , G = {G0 ; G1 ; G2 }, where Gi = {i + 3j: 06j65}, 06i62, and S6 B = r=0 Sr , where S0 = {{1; 8; 12}; {7; 14; 0}; {13; 2; 6}; {3; 10; 17}; {9; 16; 5}; {15; 4; 11}}, S1 = {{1; 9; 11}; {7; 15; 17}; {13; 3; 5}; {0; 2; 10}; {6; 8; 16}; {12; 14; 4}}, S2 = {{2; 4; 9}; {8; 10; 15}; {14; 16; 3}; {0; 1; 17}; {6; 7; 5}; {12; 13; 11}}, S3 = {{1; 2; 3}; {7; 8; 9}; {13; 14; 15}; {0; 4; 5}; {6; 10; 11}; {12; 16; 17}}, S4 = {{9; 10; 14}; {7; 11; 3}; {13; 0; 8}; {4; 6; 17}}, S5 = {{15; 16; 2}; {13; 17; 9}; {1; 6; 14}; {10; 12; 5}}, S6 = {{3; 4; 8}; {1; 5; 15}; {7; 12; 2}; {16; 0; 11}}. It is easy to check that (V; G; B) is a 3 − GDD(63 ) and the minimum distance in Sr , 06r66, is 3. So, t67. Lemma 3.15. There exists a 3 − GDD(6m ) for m = 4; 5 with the property that all blocks of such a design can be partitioned into t sets S0 ; S1 ; : : : ; St−1 such that t64 and the minimum distance in Sr , 06r6t − 1; is 3. Proof. By Lemma 1.5 there exists a GS(2; 3; 4; 2) and a GS(2; 3; 5; 3). Use these designs as master GDDs in Wilson’s Fundamental construction and give weight three or two to each element, we get a 3 − GDD(64 ) or a 3 − GDD(65 ), respectively. It is not dicult to see that the resultant designs are desired ones. By Lemmas 3.13–3.15 we have the following. Lemma 3.16. If there exists a GS(2; 3; n; 6); then there exists a GS(2; 3; mn; 6) and a GS(2; 3; m(n − 1) + 1; 6); where m = 3; 4; 5. Proof. Since a GS(2; 3; n; 6) exists, we have from Lemma 1.4 that n¿8. For m=3; 4; 5, a 3 − GDD(6m ) exists from Lemmas 3.14 –3.15 with t67. Then the conclusion comes from Lemma 3.13. Lemma 3.17. There exists a GS(2; 3; v; 6) for v ∈ E\([8; 21]∪{23; 26; 35; 38; 47; 59; 62}). Proof. For any v ∈ E \([8; 21] ∪ {23; 26; 35; 38; 47; 59; 62}), we can write v = mn or v = m(n − 1) + 1 for some m ∈ {3; 4; 5} and n ∈ [8; 21] ∪ {23; 26}. By Lemmas 3.12 and 3.16 there exists a GS(2; 3; v; 6). Here, we list the triples (v; m; n) in Table 1. To determine the existence of GS(2; 3; n; 6) for the remaining values n of E, singular indirect product (SIP) construction for GS(2; 3; n; g)s (see Chen et al. (1999), Theorem 2:3) will be used. We need some notations. A holey group divisible design, K −HGDD, is a four-tuple (V; G; H; B), where V is a set of points, G is a partition of V into
392
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
Table 1 Triples (v; m; n) for v ∈ E \([8; 21] ∪ {23; 26; 35; 38; 47; 59; 62}) v
m
n
v
m
n
v
m
n
22 = 3 · 7 + 1 27 = 3 · 9 30 = 3 · 10 33 = 3 · 11 37 = 3 · 12 + 1 41 = 4 · 10 + 1 44 = 4 · 11 48 = 3 · 16 51 = 3 · 17 54 = 3 · 18 58 = 3 · 19 + 1 63 = 3 · 21 68 = 4 · 17 71 = 5 · 14 + 1 77 = 4 · 19 + 1
3 3 3 3 3 4 4 3 3 3 3 3 4 5 4
8 9 10 11 13 11 11 16 17 18 20 21 17 15 20
24 = 3 · 8 28 = 3 · 9 + 1 31 = 3 · 10 + 1 34 = 3 · 11 + 1 39 = 3 · 13 42 = 3 · 14 45 = 3 · 15 49 = 3 · 16 + 1 52 = 3 · 17 + 1 55 = 3 · 18 + 1 60 = 3 · 20 66 = 5 · 15 + 1 69 = 3 · 23 75 = 5 · 15 78 = 3 · 26
3 3 3 3 3 3 3 3 3 3 3 5 3 5 3
8 10 11 12 13 14 15 17 18 19 20 16 23 15 26
25 = 3 · 8 + 1 29 = 4 · 7 + 1 32 = 4 · 8 36 = 3 · 12 40 = 3 · 13 + 1 43 = 3 · 14 + 1 46 = 3 · 15 + 1 50 = 5 · 10 53 = 4 · 13 + 1 56 = 4 · 14 61 = 3 · 20 + 1 67 = 3 · 22 + 1 70 = 5 · 14 76 = 4 · 19 79 = 3 · 26 + 1
3 4 4 3 3 3 3 5 4 4 3 3 5 4 3
9 8 8 2 14 15 16 10 14 14 21 23 14 19 27
subsets called groups, H ⊂ G; B is a set of blocks such that a group and a block contain at most one common point and every pair of points from distinct groups, not both in H, occurs in a unique block in B, where |B| ∈ K for any B ∈ B. A k − HGDD(g(n;u) ) denotes a K − HGDD with n groups of size g in G, u groups in H and K = {k}. A holey generalized Steiner triple system, HGS(2; 3; (n; u); g), is a 3 − HGDD(g(n;u) ) with the property that any two intersecting blocks intersect at most two common groups. It is easy to see that if u = 0 or u = 1, then a HGS(2; 3; (n + u; u); g) is just a GS(2; 3; n; g) or a GS(2; 3; n + 1; g), respectively. The following lemma is clear. Lemma 3.18 (Filling in holes). If there exists a HGS(2; 3; (n; u); g) and a GS(2; 3; u; g); then there exists a GS(2; 3; n; g). Take m = 3 and g = 6 in Theorem 2:3 in Chen et al. (1999). Let n¿7 be a prime power, then the desings (1) and (2) stated in Theorem 2:3 in Chen et al. (1999) are guaranteed by Lemma 3.2 in Chen et al. and Lemma 3.14. So, we have the following. Lemma 3.19. Let n¿7 be a prime power and u; a be integers such that 06a6u6n. Suppose the following designs exist: (1) a HGS(2; 3; (n + u; u); 6); (2) a GS(2; 3; 2a + u; 6). Then there exists a GS(2; 3; 3n + 2a + u; 6). Lemma 3.20. There exists a GS(2; 3; n; 6) for n ∈ {59; 62}. Proof. A GS(2; 3; 25; 6) with a sub GS(2; 3; 9; 6) is given in the proof of Lemma 3.17. Delete the blocks of this sub GS(2; 3; 9; 6) from the GS(2; 3; 25; 6) to get a
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
393
HGS(2; 3; (25; 9); 6). A GS(2; 3; 11; 6) is guaranteed by Lemma 3.12. By taking n = 16; u = 9 and a = 1 in Lemma 3.19, a GS(2; 3; 59; 6) is obtained. Similarly, the proof of Lemma 3.17 provides a GS(2; 3; 24; 6) with a sub GS(2; 3; 8; 6). As is done above, a GS(2; 3; (24; 8); 6) is obtained. The existence of a GS(2; 3; 14; 6) is guaranteed by Lemma 3.12. Take n = 16; u = 8 and a = 3 in Lemma 3.19 to get a GS(2; 3; 62; 6). We are now in a position to prove Theorem 1.6. Proof of Theorem 1.6. The necessity comes from Lemma 1.4. For suciency, by Lemmas 3.12, 3.17 and 3.20 we know that there exists a GS(2; 3; n; 6) for any n ∈ E. So, there exists a GS(2; 3; n; 6) for any n¿8. Consequently, there exists a (6 + 1)-ary MCWC (n; 3; 3) for any n¿8. Note added in proof (June 1999) The existence of a GS(2,3,n; g) has been solved also for g = 7; 8 in Wu et al. (1999) by di erent methods. Acknowledgements The authors would like to thank the referees for their helpful comments. Appendix A. ∗
GFS(218 ; 4): S0 = {{24; 13}; {7; 1}; {17; 20}; {5; 4}; {15; 27}; {26; 19}; {23; 31}; {28; 2}; {12; 29}; {30; 21}; {14; 6}; {35; 10}; {34; 32}; {16; 11}; {33; 25}; {22; 8}; {3; 9}}; S1 = {{34; 0}; {6; 29}; {10; 11}; {28; 4}; {31; 21}; {14; 23}; {9; 33}; {18; 24}; {17; 7}; {8; 12}; {5; 27}; {32; 2}; {22; 26}; {15; 16}; {20; 13}; {35; 3}; {25; 30}}; S2 = {{18; 23}; {7; 32}; {28; 33}; {35; 16}; {15; 17}; {12; 3}; {29; 30}; {5; 9}; {34; 1}; {6; 26}; {10; 31}; {19; 22}; {25; 27}; {0; 8}; {4; 13}; {14; 21}; {24; 11}}; S3 = {{1; 26}; {28; 31}; {18; 6}; {8; 30}; {14; 24}; {9; 23}; {15; 0}; {34; 13}; {27; 7}; {12; 25}; {29; 10}; {5; 20}; {16; 32}; {11; 4}; {35; 22}; {17; 33}; {2; 19}}: ∗ GFS(219 ; 2): S0 = {{5; 23}; {27; 17}; {9; 10}; {31; 37}; {11; 25}; {6; 33}; {34; 12}; {22; 30}; {14; 1}; {28; 2}; {15; 13}; {16; 7}; {35; 20}; {8; 3}; {21; 36}; {32; 4}; {29; 18}; {24; 26}}; S1 = {{9; 25}; {33; 36}; {13; 4}; {35; 23}; {37; 34}; {2; 22}; {3; 28}; {18; 19}; {29; 21}; {5; 12}; {0; 17}; {31; 27}; {7; 24}; {15; 8}; {32; 26}; {30; 16}; {14; 10}; {11; 6}}: ∗ GFS(223 ; 2): S0 = {{36; 6}; {14; 29}; {34; 38}; {22; 7}; {3; 41}; {30; 2}; {45; 28}; {4; 9}; {32; 35}; {18; 13}; {5; 8}; {31; 17}; {19; 12}; {16; 43}; {39; 21}; {33; 27}; {20; 26}; {1; 37}; {40; 15}; {10; 11}; {25; 42}; {24; 44}};
394
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
S1 = {{30; 8}; {5; 14}; {38; 3}; {26; 12}; {2; 4}; {13; 40}; {10; 0}; {43; 23}; {42; 34}; {32; 44}; {22; 35}; {9; 11}; {33; 29}; {18; 27}; {19; 31}; {41; 17}; {25; 36}; {37; 7}; {15; 16}; {6; 39}; {21; 28}; {45; 20}}: ∗ GFS(226 ; 4): S0 = {{50; 29}; {39; 44}; {51; 34}; {23; 41}; {40; 11}; {48; 24}; {19; 35}; {47; 25}; {2; 20}; {15; 22}; {21; 38}; {13; 32}; {10; 46}; {28; 42}; {18; 8}; {14; 27}; {49; 3}; {37; 1}; {45; 17}; {9; 6}; {43; 31}; {30; 12}; {5; 7}; {16; 33}; {36; 4}}; S1 = {{28; 6}; {2; 25}; {48; 38}; {20; 16}; {50; 26}; {46; 23}; {18; 43}; {14; 11}; {51; 3}; {39; 9}; {29; 15}; {21; 33}; {13; 22}; {19; 12}; {31; 10}; {24; 8}; {0; 45}; {5; 37}; {32; 47}; {34; 30}; {41; 4}; {42; 44}; {49; 36}; {17; 35}; {40; 7}}; S2 = {{24; 27}; {19; 38}; {49; 39}; {13; 14}; {41; 0}; {46; 6}; {11; 36}; {26; 18}; {3; 45}; {51; 8}; {1; 5}; {12; 23}; {34; 43}; {32; 9}; {17; 42}; {25; 20}; {47; 33}; {22; 7}; {44; 31}; {16; 4}; {15; 35}; {48; 21}; {50; 30}; {10; 29}; {37; 40}}; S3 = {{32; 1}; {2; 17}; {37; 28}; {25; 30}; {19; 40}; {47; 48}; {10; 21}; {46; 51}; {14; 44}; {43; 45}; {42; 31}; {27; 33}; {49; 5}; {50; 4}; {20; 6}; {13; 26}; {36; 38}; {34; 41}; {35; 0}; {11; 39}; {9; 8}; {24; 16}; {12; 18}; {15; 7}; {22; 23}}: ∗ GFS(235 ; 2): S0 = {{9; 63}; {11; 43}; {36; 52}; {67; 10}; {3; 69}; {49; 21}; {37; 39}; {13; 61}; {20; 44}{55; 2}; {66; 7}; {6; 38}; {4; 41}; {47; 48}; {1; 53}; {64; 8}; {19; 46}; {62; 33}; {58; 56}; {23; 29}; {51; 31}; {25; 16}; {57; 45}; {32; 50}; {24; 30}; {60; 65}; {12; 54}; {59; 26}; {17; 40}; {18; 28}; {34; 27}; {42; 22}; {68; 15}; {14; 5}}; S1 = {{53; 8}; {65; 46}; {68; 19}; {43; 69}; {47; 33}; {60; 30}; {42; 64}; {29; 4}; {7; 37}; {40; 41}; {50; 14}; {18; 67}; {52; 56}; {31; 39}; {59; 20}; {21; 6}; {62; 15}; {48; 9}; {22; 51}; {35; 38}; {24; 12}; {0; 55}; {25; 44}; {16; 5}; {10; 17}; {57; 11}; {2; 28}; {58; 61}; {32; 45}; {49; 54}; {63; 27}; {26; 34}; {3; 13}; {23; 66}}: ∗ GS(238 ; 4): S0 = {{67; 10}; {75; 46}; {29; 64}; {5; 15}; {50; 55}; {57; 9}; {23; 30}; {18; 51}; {31; 16}; {17; 1}; {27; 19}; {60; 73}; {62; 20}; {8; 59}; {53; 47}; {13; 32}; {33; 35}; {49; 52}; {45; 25}; {70; 34}; {28; 54}; {22; 4}; {12; 24}; {74; 14}; {3; 36}; {65; 40}; {41; 58}; {63; 42}; {37; 68}; {26; 71}; {7; 48}; {72; 2}; {69; 66}; {61; 6}; {56; 11}; {44; 21}; {43; 39}}; S1 = {{40; 11}; {48; 49}; {68; 3}; {9; 58}; {69; 46}; {5; 57}; {25; 55}; {19; 6}; {66; 72}; {8; 56}; {60; 4}; {54; 2}; {16; 24}; {38; 50}; {63; 29}; {13; 14}; {7; 21}; {15; 52}; {36; 65}; {35; 47}; {44; 26}; {71; 17}; {30; 0}; {20; 74}; {31; 61}; {28; 45}; {34; 62}; {41; 10}; {37; 27}; {33; 18}; {23; 51}; {43; 70}; {67; 64}; {75; 22}; {59; 32}; {12; 73}; {53; 42}}; S2 = {{47; 11}; {23; 26}; {34; 42}; {75; 38}; {73; 71}; {50; 46}; {57; 48}; {68; 13}; {20; 64}; {63; 74}; {45; 9}; {8; 70}; {31; 15}; {69; 25}; {35; 10}; {59; 39}; {19; 0}; {41; 4}; {5; 17}; {1; 30}; {56; 58}; {22; 65}; {61; 54}; {53; 60}; {12; 52}; {62; 72}; {24; 28}; {27; 18}; {37; 32}; {3; 16}; {51; 7}; {6; 36}; {67; 49}; {43; 21}; {66; 44}; {14; 33}; {55; 29}};
K. Chen et al. / Journal of Statistical Planning and Inference 86 (2000) 379–395
395
S3 = {{61; 70}; {26; 16}; {28; 42}; {74; 18}; {38; 64}; {49; 15}; {54; 17}; {22; 9}; {25; 7}; {62; 27}; {55; 56}; {14; 46}; {63; 39}; {11; 4}; {34; 35}; {23; 44}; {72; 29}; {30; 32}; {6; 57}; {40; 24}; {50; 67}; {21; 48}; {52; 10}; {43; 20}; {51; 66}; {58; 53}; {47; 33}; {36; 31}; {8; 60}; {0; 65}; {71; 12}; {68; 59}; {19; 45}; {37; 2}; {5; 13}; {73; 1}; {69; 75}}: ∗ GFS(247 ; 2): S0 = {{7; 21}; {30; 20}; {74; 70}; {23; 48}; {75; 40}; {26; 58}; {22; 81}; {91; 41}; {83; 55}; {59; 64}; {3; 12}; {90; 63}; {33; 52}; {61; 32}; {49; 78}; {82; 16}; {66; 6}; {44; 2}; {80; 89}; {84; 10}; {15; 67}; {11; 71}; {45; 38}; {85; 13}; {54; 28}; {57; 17}; {53; 86}; {43; 24}; {5; 42}; {27; 76}; {19; 92}; {35; 29}; {88; 31}; {62; 51}; {72; 14}; {36; 50}; {34; 4}; {1; 18}; {65; 68}; {77; 93}; {87; 60}; {69; 37}; {56; 25}; {8; 79}; {9; 73}; {39; 46}}; S1 = {{56; 54}; {43; 64}; {63; 50}; {60; 36}; {77; 52}; {41; 8}; {39; 75}; {2; 51}; {47; 4}; {29; 49}; {65; 61}; {93; 83}; {19; 37}; {88; 73}; {82; 3}; {62; 9}; {46; 84}; {44; 66}; {27; 53}; {76; 33}; {79; 80}; {57; 34}; {13; 21}; {92; 42}; {58; 10}; {87; 32}; {23; 11}; {20; 38}; {90; 35}; {45; 28}; {12; 0}; {55; 7}; {14; 15}; {78; 70}; {25; 81}; {26; 31}; {71; 69}; {18; 72}; {89; 86}; {67; 91}; {5; 68}; {17; 30}; {24; 40}; {85; 74}; {6; 59}; {16; 22}}: References Beth, T., Jungnickel, D., Lenz, H., 1986. Design Theory. Cambridge University Press, London. Bird, I.C.M., Keedwell, A.D., 1994. Design and applications of optical orthogonal codes — a survey. Bull. ICA 11, 21–44. Blake-Wilson, S., Phelps, K., 1999. Constant weight codes and group divisible design. Designs, Codes and Cryptography 16, 11–27. Brickell, E.F., Wei, V.K., 1987. Optical orthogonal codes and cyclic block designs. Congr. Numer. 58, 175–192. Chen, K., Ge, G., Zhu, L., 1999. Generalized Steiner triple systems with group size ÿve. J. Combin. Des., to appear. Chung, F.P.K., Salehi, J.A., Wei, V.K., 1989. Optical orthogonal codes: design, analysis, and applications. IEEE Trans. Inform. Theory 35, 595–604. Colbourn, C.J., Dinitz, J.H. (Eds.), 1996. The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton. Dinitz, J.H., Stinson, D.R., 1992. Room squares and related designs. in: Contemporary Design Theory: A Collection of Surveys. Wiley, New York, pp. 137–204. Etzion, T., 1997. Optimal constant weight codes over Zk and generalized designs. Discrete Math. 169, 55–82. Phelps, K., Yin, C., 1997. Generalized Steiner systems with block size three and group size g ≡ 3 (mod 6). J. Combin. Des. 5, 417–432. Phelps, K., Yin, C., 1999. Generalized Steiner systems with block size three and size four. Ars Combin., to appear. Salehi, J.A., 1989. Emerging optical code-division multiple access communications systems. IEEE Network 3–2, 31–39. Salehi, J.A., Brackett, C.A., 1989. Code-division multiple access techniques in optical ÿber networks: parts 1 and 2. IEEE Trans. Commun. 37, 824–842. Wu, D., Zhu, L., Ge, G., 1999. Generalized Steiner triple systems with group size g = 7; 8. Ars Combin., to appear. Yin, J., 1998. Some combinatorial constructions for optical orthogonal codes. Discrete Math. 185, 201–219.