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A completion of LS(2n41)
Discrete Mathematics (
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A completion of LS(2n 41 )✩ Yanxun Chang a , Lijun Ji b, *, Hao Zheng a a b
Institute of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China Department of Mathematics, Soochow University, Suzhou 215006, PR China
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Article history: Received 19 July 2016 Received in revised form 27 October 2016 Accepted 28 October 2016 Available online xxxx Keywords: Large set Group-divisible design Threshold scheme
a b s t r a c t Large sets of disjoint group-divisible designs with block size three and type 2n 41 (denoted by LS(2n 41 )) were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It has been shown that the necessary condition n ≡ 0 (mod 3) for the existence of an LS(2n 41 ) is sufficient with five possible exceptions n ∈ {12, 30, 36, 48, 144}. These five undetermined LS(2n 41 )s are shown to exist in this paper. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The investigation of large sets of GDD(2, 3, 2n + 4)s with type 2n 41 was started in 1989 by Schellenberg and Stinson [18]. Such large sets of GDD(2, 3, 2n + 4)s have applications in cryptography to the construction of perfect threshold schemes (see [6,18,19]). A group-divisible t-design (GDD) of order v and block size k denoted by GDD(t , k, v ) is a triple (X , G , B) which satisfies the following properties: (i) X is a v -element set, (ii) G is a set of non-empty subsets of X called groups which partition X , (iii) B is a set of k-subsets of X (called blocks), such that a group and a block contain at most one common point, and every t-set of points from distinct groups occurs in exactly one block. The type of a GDD is the multiset {|G| : G ∈ G }. We denote the type by 1u1 2u2 , . . . , where there are precisely ui occurrences of i, i ≥ 1. Two GDD(t , k, v )s with the same group set, say (X , G , A) and (X , G , B), are said to be disjoint if A ∩ B = ∅. A set of more than two GDD(t , k, v )s (having the same group set) are called disjoint if each pair is disjoint. It is not difficult to see that the maximum number of disjoint GDD(2, 3, ut + s)s of type t u s1 is t(u − 1) for s ≥ t. Such a collection of disjoint GDD(2, 3, ut + s)s is called a large set, and denoted by LS(t u s1 ), or LS(t u+1 ) for t = s. The existence of an LS(t n ) (also denoted by LGDD(t n )) has been investigated by many authors including Lu [15,16], Teirlinck [20,21] and others (see Refs. [5–7,18]), and finally solved by Lei [14]. Theorem 1.1 ([14]). There exists an LS(t n ) if and only if n(n − 1)t 2 ≡ 0 (mod 6) and (n − 1)t ≡ 0 (mod 2) and (t , n) ̸ = (1, 7). The existence of an LS(2n 41 ) was almost completed with only five possible exceptions. ✩ This work is supported by the NSFC under Grants 11271042 (Y. Chang), 11431003 (Y. Chang, L. Ji), and a project funded by the priority academic program development of Jiangsu higher education institutions (L. Ji). Corresponding author. E-mail addresses: [email protected] (Y. Chang), [email protected] (L. Ji).
*
http://dx.doi.org/10.1016/j.disc.2016.10.020 0012-365X/© 2016 Elsevier B.V. All rights reserved.
Please cite this article in press as: Y. Chang, et al., A completion of LS(2n 41 ), Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.020
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Theorem 1.2 ([2–5,10,18]). There exists an LS(2n 41 ) if and only if n ≡ 0 (mod 3) with five possible exceptions n ∈ {12, 30, 36, 48, 144}. In this paper, the five LS(2n 41 )s are determined. In Section 2, we describe a construction of LS(2n 41 )s from partitionable H(n + 2, 2, 3, 3) frames and give a construction of partitionable H(3q − 1, 2, 3, 3) frames from H(q, 2, 4, 3) frames. In the last section, five unsolved LS(2n 41 )s are constructed, thereby, we have the following result. Theorem 1.3. There exists an LS(2n 41 ) if and only if n ≡ 0 (mod 3). 2. A construction of LS(2n 41 )s via partitionable HF (n + 2, 2, 3, 3)s In this section, following [10] we state a construction of LS(2n 41 )s from partitionable H(n + 2, 2, 3, 3) frames. We also construct a partitionable H(3q − 1, 2, 3, 3) frame from an H(q, 2, 4, 3) frame. For non-negative integers q, g , k and t, an H(q, g , k, t) frame (as in [8]) is an ordered four-tuple (X , G , B, F ) with the following properties: (1) X is a set of qg points; (2) G = {G1 , G2 , . . . , Gq } is an equipartition of X into q groups; (3) F is a family of subsets of G called holes which is closed under intersections. Hence each hole Fi ∈ F is of the form Fi = {Gi1 , Gi2 , . . . , Gis }, and if Fi and Fj are holes then Fi ∩ Fj is also a hole. The number of groups in a hole is its size; (4) B is a set of k-element transverses (called blocks) of G with the property that every t-element transverse of G which is not a t-element transverse of some hole Fi ∈ F is contained in precisely one block, and no block contains a t-element transverse of some hole, where a transverse is a subset of X that meets each Gi in at most one point. In this paper, an H(q, g , k, t) frame is shortly denoted by HF (q, g∑ , k, t). When an HF (q, g , k, t) has ai holes of size qi + s (1 ≤ i ≤ r) which intersect in a common hole of size s and q = s + 1≤i≤r ai qi , for convenience such a design is denoted by a a a HF ((q11 q22 · · · qr r : s), g , k, t). When r = 1, it is uniform. a a a Let (X , G , A, F ) be an HF ((q11 q22 · · · qr r : s), g , 3, 3) and F0 be the common hole of size s with s ≥ 2. It is called a1 a2 a partitionable and denoted by PHF ((q1 q2 · · · qr r : s), g) if the block set A can be partitioned into Ax (x ∈ G, G ∈ G \ F0 ) and A1 , A2 , . . . , Ag(s−2) with the following two properties: (i) for each hole F of size qi + s, G ∈ F \ F0 and x ∈ G, Ax is the ∑ ∑ block set of a GDD(2, 3, 1≤j≤r gaj qj + gs) of∑ type g 1≤j≤r aj qj −qi (gqi + gs)1 with group set (G \ F ) ∪ {{x ∈ G : G ∈ F }}; (ii) for 1 ≤ i ≤ g(s − 2), (X ′ , G ′ , Ai ) is a GDD(2, 3, 1≤j≤r gaj qj ) of type (gq1 )a1 (gq2 )a2 · · · (gqr )ar , where X ′ = {x ∈ G : G ∈ G \ F0 } and G ′ = {{x ∈ G : G ∈ F \ F0 } : F ∈ F \ {F0 }}. When g = 1, such a PHF is a partitionable candelabra system with block size three (PCS) [1,11], which has been used to determine M(6k + 5) = 6k + 1 and to give a new existence proof of large sets of disjoint Steiner triple system [9]. Let (X , G , {Ai : 1 ≤ i ≤ 2(n − 1)}) be an LS(2n 41 ). If there are subsets Y ⊂ X , G ′ ⊂ G and A′i ⊂ Ai , 1 ≤ i ≤ 2(w − 1), such that (Y , G ′ , {A′i : 1 ≤ i ≤ 2(w − 1)}) is an LS(2w 41 ), then such a system is denoted by LS(2n 41 : 2w 41 ). Lemma 2.1 ([10]). Suppose there exists a PHF ((mn : s), 2). If there exists an LS(2m+s−2 41 : 2s−2 41 ), then there is an LS(2mn+s−2 41 ). A generalized frame (as in [22]) F (3, 3, n{g }) is a GDD(3, 3, ng) of type g n (X , G , A) such that the block set A can be partitioned into gn subsets Ay , y ∈ G and G ∈ G , each (X \ G, G \ {G}, Ay ) being a GDD(2, 3, g(n − 1)) of type g n−1 . Teirlinck pointed out in [22] that an F (3, 3, n{g }) can be obtained from GDD(3, 4, ng) of type g n (called an H design by Mills). Lemma 2.2 ([12,17,22]). For n > 3 and n ̸ = 5, an F (3, 3, n{g }) exists if and only if gn is even and g(n − 1)(n − 2) is divisible by 3. For n = 5, an F (3, 3, 5{g }) exists if g is even, g ̸ = 2 and g ̸ ≡ 10, 26 (mod 48). Let g ≥ 3 and let (X , G , A) be a GDD(3, 3, g(n + 1) − 1) of type g n (g − 1)1 and G0 the group of size g − 1. Such a GDD is shortly denoted by PGDD(g n (g − 1)1 ) if the block set A can be partitioned into Ax (x ∈ G, G ∈ G and G ̸ = G0 ) and A1 , . . . , Ag −3 with the following two properties: (i) each Ax is the block set of a GDD(2, 3, gn) of type g n with group set (G \ {G0 , G}) ∪ {G0 ∪ {x}}, (2) each (X \ G0 , G \ {G0 }, Ai ) is a GDD(2, 3, gn) of type g n . Lemma 2.3 ([9]). There is a PGDD(3k 21 ) for k = 3, 5. Lemma 2.4. If there is an HF ((mk : s), 2, 4, 3) with s ≥ 1, then there is a PHF (((3m)k : 3s − 1), 2). Proof. Let (X , G , T , F = {F0 , F1 , . . . , Fk }) be the given HF ((mk : s), 2, 4, 3) where F0 = {G1 , . . . , Gs } is the common hole of size s and Gs = {∞1 , ∞2 }. The desired design will be constructed on Y = ((X \ Gs )×Z3 ) ∪ S1 ∪ S2 with the group set G ′ = {G×{i} : G ∈ G \ {Gs }, i ∈ Z3 } ∪ {S1 , S2 } and the set of holes F ′ = {Fj′ : 1 ≤ j ≤ k} ∪ {F0′ }, where S1 = {∞1 }×Z2 , S2 = {∞2 }×Z2 , F0′ = {S1 , S2 } ∪ {G×{i} : G ∈ F0 \ {Gs }, i ∈ Z3 } and Fj′ = {G×{i} : i ∈ Z3 , G ∈ Fj \ F0 } ∪ F0′ for 1 ≤ j ≤ k. We describe its block set below. For l = 1, 2 and each block B ∈ T with ∞l ∈ B, construct a PGDD(33 21 ) on ((B \ {∞l })×Z3 ) ∪ Sl with groups {x}×Z3 , x ∈ B \ {∞l } and Sl . Such a design exists by Lemma 2.3. Denote its block set CBl . Then CBl can be partitioned into 9 parts Please cite this article in press as: Y. Chang, et al., A completion of LS(2n 41 ), Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.020
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CBl (x, i), (x, i) ∈ (B \ {∞l })×Z3 , each being the block set of a GDD(2, 3, 9) of type 33 with groups {y}×Z3 , y ∈ B \ {x, ∞l }, and {(x, i)} ∪ Sl . For each block B ∈ T with B ∩ Gs = ∅, construct an F (3, 3, 4{3}) on B×Z3 with the group set ΓB = {{x}×Z3 : x ∈ B}. Such a design exists by Lemma 2.2. Denote its block set by DB . Then DB can be partitioned into 12 parts DB (x, i), (x, i) ∈ B×Z3 , each being the block set of a GDD(2, 3, 9) of type 33 with the group set ΓB \ {{x}×Z3 }. For each group G ∈ G \ F0 , let FG = {FG0 , FG1 , FG2 , FG3 } be a one-factorization of the complete tripartite graph on G×Z3 with three partite sets G×{i}, i ∈ Z3 . For 1 ≤ j ≤ k, each G = {x, z } ∈ Fj \ F0 , i ∈ Z3 and G′ ∈ G \ Fj , define A(x, i, G′ ) = {{(x, i), a, b} : {a, b} ∈ FG3′ } ∪ {{(z , n), a, b} : {a, b} ∈ FG′ , n ∈ Z3 }, n+i
and
⋃
A(x, i) =
A(x, i, G′ ),
G′ ∈G \F
j
where n + i is computed modulo 3. ⋃ ⋃ ⋃ We claim that ( l=1,2,B∈T ,∞ ∈B CBl ) ∪ ( B∈T ,B∩Gs =∅ DB ) ∪ ( x∈G∈G \F ,i∈Z A(x, i)) is an HF (((3m)k : 3s − 1), 2, 3, 3). To check l 0 3 that each 3-transverse T = {(a1 , b1 ), (a2 , b2 ), (a3 , b3 )} of the group set G ′ which is not a 3-transverse of some hole of F ′ is contained in exactly one block, we consider the following two cases. (i) If |T ∩ (G′ ×Z3 )| = 2 for some group G′ ∈ Fj (1 ≤ j ≤ k), without loss of generality let {(a2 , b2 ), (a3 , b3 )} ⊂ G′ ×Z3 . Then (a1 , b1 ) is not contained in any group of Fj′ . When {(a2 , b2 ), (a3 , b3 )} ∈ FG3′ , we have that T ∈ A(a1 , b1 , G′ ) ⊂ A(a1 , b1 ). When {(a2 , b2 ), (a3 , b3 )} ∈ FGn′ for some n ∈ Z3 . Let a1 ∈ G and x be the other element in the group G. Then T ∈ A(x, n − b1 , G′ ) ⊂ A(x, n − b1 ), where n − b1 is computed modulo 3. (ii) If |T ∩ (G×Z3 )| ≤ 1 for any group G ∈ G , then {a1 , a2 , a3 } is a 3-transverse of the group set G which is not a 3-transverse of some hole of F . By the definition of an HF ((mk : s), 2, 4, 3), there is unique block B ∈ T such that {a1 , a2 , a3 } ⊂ B. When B ∩ Gs = {∞l } for some l ∈ {1, 2}, T occurs in CBl exactly once. When B ∩ Gs = ∅, T occurs in DB exactly once. Therefore, our claim holds. It is left to show that the obtained design is also partitionable. By the definition, it should contain exactly 6mk GDD(2, 3, 6mk + 6s − 2)s of type 23mk−3m (6m + 6s − 2)1 and 6s − 6 GDD(2, 3, 6mk)s of type (6m)k . For 1 ≤ j ≤ k, G ∈ Fj \ F0 , x ∈ G and i ∈ Z3 , let
⎞ ⎛ ⎞ ⎛ ⎞ ⋃ ⋃ ⋃ ⋃ ⋃ ⋃ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ DB (x, i)⎠ , A(x, i − 1) CB2 (x, i + 1)⎠ CB1 (x, i)⎠ F (x, i) = ⎝ ⎝ ⎝ ⎛
and for (y, i) ∈
(⋃
F (y, i) =
G′ ∈F0 \{Gs } G
⋃
x∈B∈T B∩Gs =∅
B∈T {x,∞2 }⊂B
B∈T {x,∞1 }⊂B
′
) ×Z3 , let
DB (y, i).
y∈B∈T
It is easy to see that all F (x, i) and F (y, i) form a partition of the block set of an HF (((3m)k : 3s − 1), 2, 3, 3). We have 1 to prove that (x, i) is the block set of a GDD(2, 3, 6mk + 6s − 2) of type 23mk−3m (6m + set ⋃ each F ⋃6s − 2) ′′with group ′ ′ ′′ k (G \ Fj ) ∪ { G′′ ∈F ′ G } and F (y, i) is the block set of a GDD(2, 3, 6mk) of type (6m) with groups G′′ ∈F ′ \F ′ G , 1 ≤ j′ ≤ k. To j′
j
0
check that each pair of points P = {(a1 , b1 ), (a2 , b2 )} not contained in any group is contained in a unique block of F (x, i), we consider the following two cases: ⋃ ⋃ ⋃ ′′ (1) If P ∩ ( G′′ ∈F ′ G′′ ) ̸ = ∅, then |P ∩ ( G′′ ∈F ′ G′′ )| = 1 and without loss of generality let (a2 , b2 ) ∈ G′′ ∈F ′ G . Then j
a1 ̸ ∈
⋃
′
G′ ∈Fj G
j
j
. When a2 = ∞l (l ∈ {1, 2}), then there is a unique block B ∈ T containing {x, a1 , ∞l }. Since CBl (x, i + l − 1)
is the block set of a GDD(2, 3, 9) of type 33 with groups {z }×Z3 , z ⋃ ∈ B \ {x, ∞l }, and {(x, i + l − 1)} ∪ Sl , there is a unique ′ block A ∈ CBl (x, i + l − 1) ⊂ F (x, i) such that P ⊂ A. When a2 ∈ G′ ∈Fj \{G,Gs } G , there is a unique block B ∈ T containing {x, a1 , a2 } and a unique block A ∈ DB (x, i) ⊂ F (x, i) containing P. When a2 = x, there is a unique block B ∈ T containing {a1 , a2 , ∞1 } and a unique block A ∈ CB1 (x, i) ⊂ F (x, i) containing P if b2 = i, a unique block B ∈ T containing {a1 , a2 , ∞2 } and a unique block A ∈ CB2 (x, i + 1) ⊂ F (x, i) containing P if b2 = i + 1, and a unique group G′ ∈ G containing a1 and a unique block A ∈ A(x, i − 1, G′ ) ⊂ A(x, i − 1) ⊂ F (x, i) containing P if b2 = i − 1. When a2 is the other element of the group G, there is a unique group G′ containing a1 and P ⊂ {(a2 , b2 ), (a1 , b1 ), (a3 , b3 )} ∈ A(x, i − 1, G′ ) ⊂ A(x, i − 1), where b2 +i−1 {(a1 , b1 ), (a3 , b⋃ and b2 + i − 1 is computed 3 )} ∈ FG′ ⋃ modulo 3. (2) If P ∩ ( G′′ ∈F ′ G′′ ) = ∅ then {a1 , a2 } ⊂ X \ ( G′ ∈G \F G′ ). When a1 and a2 are from the same group G′ ̸ = G, we have j
j
that P is in some one-factor. If P ∈ FG3′ , then P ⊂ {(x, i − 1), (a1 , b1 ), (a2 , b2 )} ∈ A(x, i − 1) ⊂ F (x, i). Otherwise, P ∈ FGn′ for some n ∈ Z3 . Let z be the other element of the group G. Then P ⊂ {(z , n + 1 − i), (a1 , b1 ), (a2 , b2 )} ∈ A(x, i − 1) ⊂ F (x, i), where n + 1 − i is computed modulo 3. When a1 and a2 are from distinct groups, there is a unique block B ∈ T containing {x, a1 , a2 }. When B ∩ Gs = {∞l } (or B ∩ Gs ̸= ∅), there is a unique block A ∈ CBl (x, i) (or A ∈ DB (x, i)) containing P. So, F (x, i) is the block set of a GDD(2, 3, 6mk + 6s − 2) of type 23mk−3m (6m + 6s − 2)1 . Please cite this article in press as: Y. Chang, et al., A completion of LS(2n 41 ), Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.020
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For each pair P = {(a1 , b1 ), (a2 , b2 )} from two distinct groups
)
– ′′
⋃
G′′ ∈F ′′ \F0′ G
, 1 ≤ j′ ≤ k, there is a unique block B ∈ T
j
containing {a1 , a2 , y}, thereby, there is a unique block A ∈ DB (y, i) ⊂ F (y, i) containing P. So each F (y, i) is the block set of a GDD(2, 3, 6mk) of type (6m)k and the resultant design is a PHF (((3m)k : 3s − 1), 2). This completes the proof. □ 3. Proof of the main result Lemma 3.1. There exists an LS(212 41 : 23 41 ). Proof. Let X = Z18 ∪ Y and H = {{∞1 , ∞2 }, {∞3 , ∞4 }, {∞5 , ∞6 }, {∞7 , ∞8 , ∞9 , ∞10 }} where Y = {∞i : 1 ≤ i ≤ 10}. By Theorem 1.2 there exists an LS(23 41 ) (Y , H, {Dj : j = 1, 2, 3, 4}). We will construct an LS(212 41 : 23 41 ) on X with group set G = {{i, i + 9} : 0 ≤ i ≤ 8} ∪ H. Let A0 consist of the following triples:
{0, 1, 2} {13, ∞1 , ∞4 } {9, 14, ∞4 } {5, 13, ∞6 } {17, ∞4 , ∞6 } {0, 7, 12} {2, ∞3 , ∞10 } {3, 4, 16} {6, 10, 13} {9, 13, ∞7 } {6, 14, ∞8 } {11, 14, ∞9 } {11, 17, ∞1 } {3, 15, ∞9 } {3, ∞1 , ∞7 } {9, 15, ∞6 } {1, 9, 11} {12, ∞5 , ∞7 } {4, 10, 12} {4, 11, ∞7 } {1, ∞3 , ∞8 } {2, 3, 5} {14, ∞2 , ∞6 } {8, ∞2 , ∞4 }
{5, 7, ∞9 } {2, 7, 15} {0, 3, 17} {0, 4, ∞10 } {2, 16, ∞1 } {17, ∞5 , ∞9 } {0, 10, 14} {8, 14, ∞5 } {0, 15, ∞3 } {0, ∞1 , ∞6 } {3, 9, ∞10 } {3, 11, ∞4 } {7, ∞4 , ∞10 } {1, 4, 7} {3, ∞2 , ∞3 } {12, 15, ∞4 } {4, 6, ∞6 } {6, ∞1 , ∞3 } {1, 15, ∞5 } {1, 17, ∞7 } {8, 12, ∞9 } {8, 15, 16} {14, ∞1 , ∞10 } {14, ∞3 , ∞7 }
{9, 10, ∞9 } {2, 9, ∞2 } {2, 12, 14} {0, 5, 8} {5, 17, ∞8 } {10, 11, ∞6 } {2, ∞5 , ∞8 } {10, 15, ∞1 } {10, ∞3 , ∞5 } {3, 8, ∞6 } {0, ∞4 , ∞5 } {11, 15, ∞8 } {1, 3, 14} {7, 8, ∞7 } {7, 10, 17} {7, 13, 14} {1, 12, 13} {12, ∞6 , ∞8 } {13, 16, ∞10 } {8, 10, ∞10 } {1, ∞4 , ∞9 } {4, ∞1 , ∞9 } {13, ∞2 , ∞8 } {3, 6, 7}
{15, ∞2 , ∞7 } {5, 10, ∞2 } {5, 12, 16} {16, ∞6 , ∞7 } {0, 6, ∞7 } {9, 12, ∞1 } {0, 11, 13} {6, 9, 17} {0, 16, ∞8 } {0, ∞2 , ∞9 } {6, 16, ∞9 } {3, 13, ∞5 } {2, 10, ∞7 } {1, 5, 6} {2, ∞6 , ∞9 } {1, 8, ∞1 } {7, ∞1 , ∞8 } {7, ∞3 , ∞6 } {8, 9, ∞8 } {4, 14, 15} {1, ∞6 , ∞10 } {2, 6, ∞4 } {2, 4, 17} {5, 11, ∞3 }
{2, 8, 13} {17, ∞2 , ∞10 } {9, 16, ∞3 } {5, 15, ∞10 } {5, ∞1 , ∞5 } {5, ∞4 , ∞7 } {6, 8, 11} {10, 16, ∞4 } {6, 12, ∞2 } {11, 12, ∞10 } {3, 10, ∞8 } {11, 16, ∞5 } {6, ∞5 , ∞10 } {7, 9, ∞5 } {7, 11, ∞2 } {12, 17, ∞3 } {4, 8, ∞3 } {13, 15, 17} {1, 16, ∞2 } {13, ∞3 , ∞9 } {14, 16, 17} {4, ∞2 , ∞5 } {4, ∞4 , ∞8 } {4, 5, 9}
Let C1 be the set consisting of the following base triples:
{0, 8, 12} {∞7 , 0, 5}
{1, 5, 7} {∞8 , 1, 6}
{∞4 , 1, 2} {∞9 , 0, 7}
{∞5 , 0, 3} {∞10 , 1, 8}
{∞6 , 1, 4}
Let C2 be the set consisting of the following base triples:
{0, 8, 14} {∞7 , 0, 3}
{1, 3, 7} {∞8 , 1, 4} ⋃
{∞4 , 0, 7} {∞9 , 0, 5}
{∞5 , 1, 2} {∞10 , 1, 6}
{∞6 , 1, 8}
The 27 unordered pairs 0≤i≤8 {{2i, 2i + 1}, {2i, 2i + 2}, {2i + 1, 2i + 9}} can be partitioned into three matching Pj on Z18 (j = 1, 2, 3), where each Pj is listed as follows: P1 = i∈{0,6,12} {{i, i + 2}, {i + 1, i + 9}, {i + 4, i + 5}}, P2 = {{x + 2, y + 2} : {x, y} ∈ P1 } and P3 = {{x + 4, y + 4} : {x, y} ∈ P1 }.
⋃
For each i ∈ Z18 , let Ai = {{x + i, y + i, z + i} : {x, y, z } ∈ A0 }.
Define dev (Cj ) = {{x + 2a, y + 2a, z + 2a} : {x, y, z } ∈ Cj }, dev (Cj+2 ) = {{x + 1, y + 1, z + 1} : {x, y, z } ∈ dev (Cj )} for j = 1, 2. Define Bj , j = 1, 2, 3, 4, as follows:
(
) 3 ⋃ B1 = dev (C1 ) ∪ D1 ∪ {{∞k , x, y} : {x, y} ∈ Pk } ; k=1
(
) 2 ⋃ B2 = dev (C2 ) ∪ D2 ∪ {{∞k , x, y} : {x, y} ∈ Pk+1 } ∪ ({{∞3 , x, y} : {x, y} ∈ P1 }); k=1
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( B3 = dev (C3 ) ∪ D3 ∪
)
–
5
)
3 ⋃ {{∞k , x + 1, y + 1} : {x, y} ∈ Pk } ; k=1
(
) 2 ⋃ B4 = dev (C4 ) ∪ D4 ∪ {{∞k , x + 1, y + 1} : {x, y} ∈ Pk+1 } ∪ {{∞3 , x + 1, y + 1} : {x, y} ∈ P1 }. k=1
It is readily checked that each of (X , G , Ai ) (i ∈ Z18 ) and (X , G , Bj ) (j = 1, 2, 3, 4) is a GDD(2, 3, 28) of type 212 41 . Furthermore, we can check that these 22 block sets of GDDs are pairwise disjoint. Therefore, an LS(212 41 ) exists. Obviously, this LS(212 41 ) contains an LS(23 41 ) (Y , H, {Dj : j = 1, 2, 3, 4}). □ Lemma 3.2 ([5]). Suppose there exists an LS(2n 41 ), where n ̸ = 6. Then there exists an LS(23n 41 ). Lemma 3.3. There is an LS(2n 41 ) for n ∈ {30, 36, 48, 144}. Proof. For n = 36, it exists by Lemma 3.2 since there is an LS(212 41 ) by Lemma 3.1. For n = 30, start with an HF ((33 : 2), 2, 4, 3), which exists in [8, Design 1.B]. Applying Lemma 2.4 gives a PHF ((93 : 5), 2). Since there is an LS(212 41 ) containing an LS(23 41 ) by Lemma 3.1, applying Lemma 2.1 yields an LS(2n 41 ). For n = 48, from the proof of [13, Lemma 4.2], we can obtain an HF ((35 : 2), 2, 4, 3) by deleting the blocks generated by {0, i, 2i, 15 + i}, i ∈ {1, 2, 3, 4, 6, 7} and adjoining four new points to each of four subdesigns GDD(2, 3, 30)s of type 65 , respectively. For completeness, we also list the base blocks which generate the required blocks of an HF ((35 : 2), 2, 4, 3) on Z30 ∪ {x, y, z , w} with groups Gi = {i, i + 15}, 0 ≤ i < 15, G15 = {x, y} and G16 = {z , w} and holes Hj = {Gj , Gj+5 , Gj+10 , G15 , G16 }, 0 ≤ j < 5 and H5 = {G15 , G16 }, under the action of the automorphism (0 1 2 · · · 29)(x y)(z w).
{x, 0, 1, 2} {x, 1, 7, 24} {z , 1, 5, 23} {0, 1, 5, 24} {0, 1, 25, 28} {0, 3, 10, 13}
{x, 1, 3, 15} {z , 0, 1, 8} {z , 0, 7, 16} {0, 1, 6, 12} {0, 2, 4, 9} {0, 3, 12, 22}
{x, 0, 3, 11} {z , 1, 2, 20} {0, 1, 4, 18} {0, 1, 10, 23} {0, 2, 5, 16} {0, 4, 10, 17}
{x, 1, 4, 22} {z , 0, 2, 26} {1, 2, 8, 10} {0, 1, 11, 13} {0, 2, 6, 10} {0, 4, 12, 20}
{x, 0, 4, 21} {z , 1, 3, 14} {1, 3, 9, 22} {0, 1, 14, 20} {0, 2, 11, 22} {0, 5, 12, 19}
{x, 1, 5, 12} {z , 0, 3, 9} {0, 3, 7, 19} {0, 1, 17, 21} {0, 2, 12, 18} {0, 5, 13, 18}
{x, 0, 6, 22} {z , 1, 4, 17} {0, 1, 3, 26} {0, 1, 22, 27} {0, 3, 8, 24} {0, 5, 14, 23}
Applying Lemma 2.4 gives a PHF ((95 : 5), 2). Since there is an LS(212 41 ) containing an LS(23 41 ) by Lemma 3.1, applying Lemma 2.1 yields an LS(2n 41 ). For n = 144, it can be obtained by Lemma 3.2 with the known LS(248 41 ). □ By Lemmas 3.1, 3.3 and Theorem 1.2, the existence of an LS(2n 41 ) is completely determined, i.e., Theorem 1.3 holds. From the definition of an LS(2n 41 ) and simple computation, the number of triples from distinct groups that do not occur 4n(n−1) 4n(n−1) . It is easy to see that these triples cover every pair of points from two in any GDD(2, 3, 2n + 4) of type 2n 41 is 3 3
distinct groups of size 2 twice, i.e., these triples form the block set of a GDD(2, 3, 2n) of type 2n with index 2. It is 3 4n(n−1) interesting if these triples can be partitioned into two parts, each being the block set of a GDD(2, 3, 2n) of type 2n . If 3 n 1 they do, then this LS(2 4 ) is denoted by LS + (2n 41 ). For example, for n = 3, let {F0 , F1 , F2 , F3 } be a one-factorization of complete tripartite graph on Z3 ×Z2 with three partite sets {i}×Z2 , i ∈ {0, 1, 2}. Denote S = {∞0 , ∞1 , ∞2 , ∞3 }. Then, for i ∈ Z4 , each {{∞i+j , a, b} : {a, b} ∈ Fj , j ∈ Z4 } is the block set of GDD(2, 3, 10) of type 23 41 , they form an LS(23 41 ) on (Z3 ×Z2 ) ∪ S with group set {{k}×Z2 : k ∈ Z3 } ∪ {S }, and each {{(0, a0 ), (1, a1 ), (2, a2 )} : a0 , a1 , a2 ∈ Z2 , a0 + a1 + a2 ≡ l (mod 2)}, l ∈ Z2 , is the block set of a GDD(2, 3, 6) of type 23 with group set {{k}×Z2 : k ∈ Z3 }. So, there is an LS + (23 41 ). From the construction of an LS(2mn+s−2 41 ) from a PHF ((mn : s), 2), one can obtain an LS + (2mn+s−2 41 ) if there is an LS + (2m+s−2 41 : 2s−2 41 ). We end this paper with the following conjecture. 4n(n−1)
Conjecture. There is an LS + (2n 41 ) for any n ≡ 0 (mod 3). Acknowledgments The authors would like to thank the reviewers for many comments and Prof. L. Zhu for helpful suggestions on this topic. References [1] H. Cao, L. Ji, L. Zhu, Large sets of disjoint packings on 6k + 5 points, J. Combin. Theory (A) 108 (2004) 169–183. [2] H. Cao, J. Lei, L. Zhu, Large sets of disjoint group-divisible designs with block size three and type 2n 41 , J. Combin. Des. 9 (2001) 285–296. [3] H. Cao, J. Lei, L. Zhu, Further results on large sets of disjoint group-divisible designs with block size three and type 2n 41 , J. Combin. Des. 11 (2003) 24–35. [4] H. Cao, J. Lei, L. Zhu, Constructions of large sets of disjoint group-divisible designs LS(2n 41 ) using a generalization of *LS(2n ), Discrete Math. 338 (2015) 1449–1459.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
A completion of LS(2n 41 )✩ Yanxun Chang a , Lijun Ji b, *, Hao Zheng a a b
Institute of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China Department of Mathematics, Soochow University, Suzhou 215006, PR China
article
info
Article history: Received 19 July 2016 Received in revised form 27 October 2016 Accepted 28 October 2016 Available online xxxx Keywords: Large set Group-divisible design Threshold scheme
a b s t r a c t Large sets of disjoint group-divisible designs with block size three and type 2n 41 (denoted by LS(2n 41 )) were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It has been shown that the necessary condition n ≡ 0 (mod 3) for the existence of an LS(2n 41 ) is sufficient with five possible exceptions n ∈ {12, 30, 36, 48, 144}. These five undetermined LS(2n 41 )s are shown to exist in this paper. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The investigation of large sets of GDD(2, 3, 2n + 4)s with type 2n 41 was started in 1989 by Schellenberg and Stinson [18]. Such large sets of GDD(2, 3, 2n + 4)s have applications in cryptography to the construction of perfect threshold schemes (see [6,18,19]). A group-divisible t-design (GDD) of order v and block size k denoted by GDD(t , k, v ) is a triple (X , G , B) which satisfies the following properties: (i) X is a v -element set, (ii) G is a set of non-empty subsets of X called groups which partition X , (iii) B is a set of k-subsets of X (called blocks), such that a group and a block contain at most one common point, and every t-set of points from distinct groups occurs in exactly one block. The type of a GDD is the multiset {|G| : G ∈ G }. We denote the type by 1u1 2u2 , . . . , where there are precisely ui occurrences of i, i ≥ 1. Two GDD(t , k, v )s with the same group set, say (X , G , A) and (X , G , B), are said to be disjoint if A ∩ B = ∅. A set of more than two GDD(t , k, v )s (having the same group set) are called disjoint if each pair is disjoint. It is not difficult to see that the maximum number of disjoint GDD(2, 3, ut + s)s of type t u s1 is t(u − 1) for s ≥ t. Such a collection of disjoint GDD(2, 3, ut + s)s is called a large set, and denoted by LS(t u s1 ), or LS(t u+1 ) for t = s. The existence of an LS(t n ) (also denoted by LGDD(t n )) has been investigated by many authors including Lu [15,16], Teirlinck [20,21] and others (see Refs. [5–7,18]), and finally solved by Lei [14]. Theorem 1.1 ([14]). There exists an LS(t n ) if and only if n(n − 1)t 2 ≡ 0 (mod 6) and (n − 1)t ≡ 0 (mod 2) and (t , n) ̸ = (1, 7). The existence of an LS(2n 41 ) was almost completed with only five possible exceptions. ✩ This work is supported by the NSFC under Grants 11271042 (Y. Chang), 11431003 (Y. Chang, L. Ji), and a project funded by the priority academic program development of Jiangsu higher education institutions (L. Ji). Corresponding author. E-mail addresses: [email protected] (Y. Chang), [email protected] (L. Ji).
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http://dx.doi.org/10.1016/j.disc.2016.10.020 0012-365X/© 2016 Elsevier B.V. All rights reserved.
Please cite this article in press as: Y. Chang, et al., A completion of LS(2n 41 ), Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.020
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Theorem 1.2 ([2–5,10,18]). There exists an LS(2n 41 ) if and only if n ≡ 0 (mod 3) with five possible exceptions n ∈ {12, 30, 36, 48, 144}. In this paper, the five LS(2n 41 )s are determined. In Section 2, we describe a construction of LS(2n 41 )s from partitionable H(n + 2, 2, 3, 3) frames and give a construction of partitionable H(3q − 1, 2, 3, 3) frames from H(q, 2, 4, 3) frames. In the last section, five unsolved LS(2n 41 )s are constructed, thereby, we have the following result. Theorem 1.3. There exists an LS(2n 41 ) if and only if n ≡ 0 (mod 3). 2. A construction of LS(2n 41 )s via partitionable HF (n + 2, 2, 3, 3)s In this section, following [10] we state a construction of LS(2n 41 )s from partitionable H(n + 2, 2, 3, 3) frames. We also construct a partitionable H(3q − 1, 2, 3, 3) frame from an H(q, 2, 4, 3) frame. For non-negative integers q, g , k and t, an H(q, g , k, t) frame (as in [8]) is an ordered four-tuple (X , G , B, F ) with the following properties: (1) X is a set of qg points; (2) G = {G1 , G2 , . . . , Gq } is an equipartition of X into q groups; (3) F is a family of subsets of G called holes which is closed under intersections. Hence each hole Fi ∈ F is of the form Fi = {Gi1 , Gi2 , . . . , Gis }, and if Fi and Fj are holes then Fi ∩ Fj is also a hole. The number of groups in a hole is its size; (4) B is a set of k-element transverses (called blocks) of G with the property that every t-element transverse of G which is not a t-element transverse of some hole Fi ∈ F is contained in precisely one block, and no block contains a t-element transverse of some hole, where a transverse is a subset of X that meets each Gi in at most one point. In this paper, an H(q, g , k, t) frame is shortly denoted by HF (q, g∑ , k, t). When an HF (q, g , k, t) has ai holes of size qi + s (1 ≤ i ≤ r) which intersect in a common hole of size s and q = s + 1≤i≤r ai qi , for convenience such a design is denoted by a a a HF ((q11 q22 · · · qr r : s), g , k, t). When r = 1, it is uniform. a a a Let (X , G , A, F ) be an HF ((q11 q22 · · · qr r : s), g , 3, 3) and F0 be the common hole of size s with s ≥ 2. It is called a1 a2 a partitionable and denoted by PHF ((q1 q2 · · · qr r : s), g) if the block set A can be partitioned into Ax (x ∈ G, G ∈ G \ F0 ) and A1 , A2 , . . . , Ag(s−2) with the following two properties: (i) for each hole F of size qi + s, G ∈ F \ F0 and x ∈ G, Ax is the ∑ ∑ block set of a GDD(2, 3, 1≤j≤r gaj qj + gs) of∑ type g 1≤j≤r aj qj −qi (gqi + gs)1 with group set (G \ F ) ∪ {{x ∈ G : G ∈ F }}; (ii) for 1 ≤ i ≤ g(s − 2), (X ′ , G ′ , Ai ) is a GDD(2, 3, 1≤j≤r gaj qj ) of type (gq1 )a1 (gq2 )a2 · · · (gqr )ar , where X ′ = {x ∈ G : G ∈ G \ F0 } and G ′ = {{x ∈ G : G ∈ F \ F0 } : F ∈ F \ {F0 }}. When g = 1, such a PHF is a partitionable candelabra system with block size three (PCS) [1,11], which has been used to determine M(6k + 5) = 6k + 1 and to give a new existence proof of large sets of disjoint Steiner triple system [9]. Let (X , G , {Ai : 1 ≤ i ≤ 2(n − 1)}) be an LS(2n 41 ). If there are subsets Y ⊂ X , G ′ ⊂ G and A′i ⊂ Ai , 1 ≤ i ≤ 2(w − 1), such that (Y , G ′ , {A′i : 1 ≤ i ≤ 2(w − 1)}) is an LS(2w 41 ), then such a system is denoted by LS(2n 41 : 2w 41 ). Lemma 2.1 ([10]). Suppose there exists a PHF ((mn : s), 2). If there exists an LS(2m+s−2 41 : 2s−2 41 ), then there is an LS(2mn+s−2 41 ). A generalized frame (as in [22]) F (3, 3, n{g }) is a GDD(3, 3, ng) of type g n (X , G , A) such that the block set A can be partitioned into gn subsets Ay , y ∈ G and G ∈ G , each (X \ G, G \ {G}, Ay ) being a GDD(2, 3, g(n − 1)) of type g n−1 . Teirlinck pointed out in [22] that an F (3, 3, n{g }) can be obtained from GDD(3, 4, ng) of type g n (called an H design by Mills). Lemma 2.2 ([12,17,22]). For n > 3 and n ̸ = 5, an F (3, 3, n{g }) exists if and only if gn is even and g(n − 1)(n − 2) is divisible by 3. For n = 5, an F (3, 3, 5{g }) exists if g is even, g ̸ = 2 and g ̸ ≡ 10, 26 (mod 48). Let g ≥ 3 and let (X , G , A) be a GDD(3, 3, g(n + 1) − 1) of type g n (g − 1)1 and G0 the group of size g − 1. Such a GDD is shortly denoted by PGDD(g n (g − 1)1 ) if the block set A can be partitioned into Ax (x ∈ G, G ∈ G and G ̸ = G0 ) and A1 , . . . , Ag −3 with the following two properties: (i) each Ax is the block set of a GDD(2, 3, gn) of type g n with group set (G \ {G0 , G}) ∪ {G0 ∪ {x}}, (2) each (X \ G0 , G \ {G0 }, Ai ) is a GDD(2, 3, gn) of type g n . Lemma 2.3 ([9]). There is a PGDD(3k 21 ) for k = 3, 5. Lemma 2.4. If there is an HF ((mk : s), 2, 4, 3) with s ≥ 1, then there is a PHF (((3m)k : 3s − 1), 2). Proof. Let (X , G , T , F = {F0 , F1 , . . . , Fk }) be the given HF ((mk : s), 2, 4, 3) where F0 = {G1 , . . . , Gs } is the common hole of size s and Gs = {∞1 , ∞2 }. The desired design will be constructed on Y = ((X \ Gs )×Z3 ) ∪ S1 ∪ S2 with the group set G ′ = {G×{i} : G ∈ G \ {Gs }, i ∈ Z3 } ∪ {S1 , S2 } and the set of holes F ′ = {Fj′ : 1 ≤ j ≤ k} ∪ {F0′ }, where S1 = {∞1 }×Z2 , S2 = {∞2 }×Z2 , F0′ = {S1 , S2 } ∪ {G×{i} : G ∈ F0 \ {Gs }, i ∈ Z3 } and Fj′ = {G×{i} : i ∈ Z3 , G ∈ Fj \ F0 } ∪ F0′ for 1 ≤ j ≤ k. We describe its block set below. For l = 1, 2 and each block B ∈ T with ∞l ∈ B, construct a PGDD(33 21 ) on ((B \ {∞l })×Z3 ) ∪ Sl with groups {x}×Z3 , x ∈ B \ {∞l } and Sl . Such a design exists by Lemma 2.3. Denote its block set CBl . Then CBl can be partitioned into 9 parts Please cite this article in press as: Y. Chang, et al., A completion of LS(2n 41 ), Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.020
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CBl (x, i), (x, i) ∈ (B \ {∞l })×Z3 , each being the block set of a GDD(2, 3, 9) of type 33 with groups {y}×Z3 , y ∈ B \ {x, ∞l }, and {(x, i)} ∪ Sl . For each block B ∈ T with B ∩ Gs = ∅, construct an F (3, 3, 4{3}) on B×Z3 with the group set ΓB = {{x}×Z3 : x ∈ B}. Such a design exists by Lemma 2.2. Denote its block set by DB . Then DB can be partitioned into 12 parts DB (x, i), (x, i) ∈ B×Z3 , each being the block set of a GDD(2, 3, 9) of type 33 with the group set ΓB \ {{x}×Z3 }. For each group G ∈ G \ F0 , let FG = {FG0 , FG1 , FG2 , FG3 } be a one-factorization of the complete tripartite graph on G×Z3 with three partite sets G×{i}, i ∈ Z3 . For 1 ≤ j ≤ k, each G = {x, z } ∈ Fj \ F0 , i ∈ Z3 and G′ ∈ G \ Fj , define A(x, i, G′ ) = {{(x, i), a, b} : {a, b} ∈ FG3′ } ∪ {{(z , n), a, b} : {a, b} ∈ FG′ , n ∈ Z3 }, n+i
and
⋃
A(x, i) =
A(x, i, G′ ),
G′ ∈G \F
j
where n + i is computed modulo 3. ⋃ ⋃ ⋃ We claim that ( l=1,2,B∈T ,∞ ∈B CBl ) ∪ ( B∈T ,B∩Gs =∅ DB ) ∪ ( x∈G∈G \F ,i∈Z A(x, i)) is an HF (((3m)k : 3s − 1), 2, 3, 3). To check l 0 3 that each 3-transverse T = {(a1 , b1 ), (a2 , b2 ), (a3 , b3 )} of the group set G ′ which is not a 3-transverse of some hole of F ′ is contained in exactly one block, we consider the following two cases. (i) If |T ∩ (G′ ×Z3 )| = 2 for some group G′ ∈ Fj (1 ≤ j ≤ k), without loss of generality let {(a2 , b2 ), (a3 , b3 )} ⊂ G′ ×Z3 . Then (a1 , b1 ) is not contained in any group of Fj′ . When {(a2 , b2 ), (a3 , b3 )} ∈ FG3′ , we have that T ∈ A(a1 , b1 , G′ ) ⊂ A(a1 , b1 ). When {(a2 , b2 ), (a3 , b3 )} ∈ FGn′ for some n ∈ Z3 . Let a1 ∈ G and x be the other element in the group G. Then T ∈ A(x, n − b1 , G′ ) ⊂ A(x, n − b1 ), where n − b1 is computed modulo 3. (ii) If |T ∩ (G×Z3 )| ≤ 1 for any group G ∈ G , then {a1 , a2 , a3 } is a 3-transverse of the group set G which is not a 3-transverse of some hole of F . By the definition of an HF ((mk : s), 2, 4, 3), there is unique block B ∈ T such that {a1 , a2 , a3 } ⊂ B. When B ∩ Gs = {∞l } for some l ∈ {1, 2}, T occurs in CBl exactly once. When B ∩ Gs = ∅, T occurs in DB exactly once. Therefore, our claim holds. It is left to show that the obtained design is also partitionable. By the definition, it should contain exactly 6mk GDD(2, 3, 6mk + 6s − 2)s of type 23mk−3m (6m + 6s − 2)1 and 6s − 6 GDD(2, 3, 6mk)s of type (6m)k . For 1 ≤ j ≤ k, G ∈ Fj \ F0 , x ∈ G and i ∈ Z3 , let
⎞ ⎛ ⎞ ⎛ ⎞ ⋃ ⋃ ⋃ ⋃ ⋃ ⋃ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ DB (x, i)⎠ , A(x, i − 1) CB2 (x, i + 1)⎠ CB1 (x, i)⎠ F (x, i) = ⎝ ⎝ ⎝ ⎛
and for (y, i) ∈
(⋃
F (y, i) =
G′ ∈F0 \{Gs } G
⋃
x∈B∈T B∩Gs =∅
B∈T {x,∞2 }⊂B
B∈T {x,∞1 }⊂B
′
) ×Z3 , let
DB (y, i).
y∈B∈T
It is easy to see that all F (x, i) and F (y, i) form a partition of the block set of an HF (((3m)k : 3s − 1), 2, 3, 3). We have 1 to prove that (x, i) is the block set of a GDD(2, 3, 6mk + 6s − 2) of type 23mk−3m (6m + set ⋃ each F ⋃6s − 2) ′′with group ′ ′ ′′ k (G \ Fj ) ∪ { G′′ ∈F ′ G } and F (y, i) is the block set of a GDD(2, 3, 6mk) of type (6m) with groups G′′ ∈F ′ \F ′ G , 1 ≤ j′ ≤ k. To j′
j
0
check that each pair of points P = {(a1 , b1 ), (a2 , b2 )} not contained in any group is contained in a unique block of F (x, i), we consider the following two cases: ⋃ ⋃ ⋃ ′′ (1) If P ∩ ( G′′ ∈F ′ G′′ ) ̸ = ∅, then |P ∩ ( G′′ ∈F ′ G′′ )| = 1 and without loss of generality let (a2 , b2 ) ∈ G′′ ∈F ′ G . Then j
a1 ̸ ∈
⋃
′
G′ ∈Fj G
j
j
. When a2 = ∞l (l ∈ {1, 2}), then there is a unique block B ∈ T containing {x, a1 , ∞l }. Since CBl (x, i + l − 1)
is the block set of a GDD(2, 3, 9) of type 33 with groups {z }×Z3 , z ⋃ ∈ B \ {x, ∞l }, and {(x, i + l − 1)} ∪ Sl , there is a unique ′ block A ∈ CBl (x, i + l − 1) ⊂ F (x, i) such that P ⊂ A. When a2 ∈ G′ ∈Fj \{G,Gs } G , there is a unique block B ∈ T containing {x, a1 , a2 } and a unique block A ∈ DB (x, i) ⊂ F (x, i) containing P. When a2 = x, there is a unique block B ∈ T containing {a1 , a2 , ∞1 } and a unique block A ∈ CB1 (x, i) ⊂ F (x, i) containing P if b2 = i, a unique block B ∈ T containing {a1 , a2 , ∞2 } and a unique block A ∈ CB2 (x, i + 1) ⊂ F (x, i) containing P if b2 = i + 1, and a unique group G′ ∈ G containing a1 and a unique block A ∈ A(x, i − 1, G′ ) ⊂ A(x, i − 1) ⊂ F (x, i) containing P if b2 = i − 1. When a2 is the other element of the group G, there is a unique group G′ containing a1 and P ⊂ {(a2 , b2 ), (a1 , b1 ), (a3 , b3 )} ∈ A(x, i − 1, G′ ) ⊂ A(x, i − 1), where b2 +i−1 {(a1 , b1 ), (a3 , b⋃ and b2 + i − 1 is computed 3 )} ∈ FG′ ⋃ modulo 3. (2) If P ∩ ( G′′ ∈F ′ G′′ ) = ∅ then {a1 , a2 } ⊂ X \ ( G′ ∈G \F G′ ). When a1 and a2 are from the same group G′ ̸ = G, we have j
j
that P is in some one-factor. If P ∈ FG3′ , then P ⊂ {(x, i − 1), (a1 , b1 ), (a2 , b2 )} ∈ A(x, i − 1) ⊂ F (x, i). Otherwise, P ∈ FGn′ for some n ∈ Z3 . Let z be the other element of the group G. Then P ⊂ {(z , n + 1 − i), (a1 , b1 ), (a2 , b2 )} ∈ A(x, i − 1) ⊂ F (x, i), where n + 1 − i is computed modulo 3. When a1 and a2 are from distinct groups, there is a unique block B ∈ T containing {x, a1 , a2 }. When B ∩ Gs = {∞l } (or B ∩ Gs ̸= ∅), there is a unique block A ∈ CBl (x, i) (or A ∈ DB (x, i)) containing P. So, F (x, i) is the block set of a GDD(2, 3, 6mk + 6s − 2) of type 23mk−3m (6m + 6s − 2)1 . Please cite this article in press as: Y. Chang, et al., A completion of LS(2n 41 ), Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.020
4
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For each pair P = {(a1 , b1 ), (a2 , b2 )} from two distinct groups
)
– ′′
⋃
G′′ ∈F ′′ \F0′ G
, 1 ≤ j′ ≤ k, there is a unique block B ∈ T
j
containing {a1 , a2 , y}, thereby, there is a unique block A ∈ DB (y, i) ⊂ F (y, i) containing P. So each F (y, i) is the block set of a GDD(2, 3, 6mk) of type (6m)k and the resultant design is a PHF (((3m)k : 3s − 1), 2). This completes the proof. □ 3. Proof of the main result Lemma 3.1. There exists an LS(212 41 : 23 41 ). Proof. Let X = Z18 ∪ Y and H = {{∞1 , ∞2 }, {∞3 , ∞4 }, {∞5 , ∞6 }, {∞7 , ∞8 , ∞9 , ∞10 }} where Y = {∞i : 1 ≤ i ≤ 10}. By Theorem 1.2 there exists an LS(23 41 ) (Y , H, {Dj : j = 1, 2, 3, 4}). We will construct an LS(212 41 : 23 41 ) on X with group set G = {{i, i + 9} : 0 ≤ i ≤ 8} ∪ H. Let A0 consist of the following triples:
{0, 1, 2} {13, ∞1 , ∞4 } {9, 14, ∞4 } {5, 13, ∞6 } {17, ∞4 , ∞6 } {0, 7, 12} {2, ∞3 , ∞10 } {3, 4, 16} {6, 10, 13} {9, 13, ∞7 } {6, 14, ∞8 } {11, 14, ∞9 } {11, 17, ∞1 } {3, 15, ∞9 } {3, ∞1 , ∞7 } {9, 15, ∞6 } {1, 9, 11} {12, ∞5 , ∞7 } {4, 10, 12} {4, 11, ∞7 } {1, ∞3 , ∞8 } {2, 3, 5} {14, ∞2 , ∞6 } {8, ∞2 , ∞4 }
{5, 7, ∞9 } {2, 7, 15} {0, 3, 17} {0, 4, ∞10 } {2, 16, ∞1 } {17, ∞5 , ∞9 } {0, 10, 14} {8, 14, ∞5 } {0, 15, ∞3 } {0, ∞1 , ∞6 } {3, 9, ∞10 } {3, 11, ∞4 } {7, ∞4 , ∞10 } {1, 4, 7} {3, ∞2 , ∞3 } {12, 15, ∞4 } {4, 6, ∞6 } {6, ∞1 , ∞3 } {1, 15, ∞5 } {1, 17, ∞7 } {8, 12, ∞9 } {8, 15, 16} {14, ∞1 , ∞10 } {14, ∞3 , ∞7 }
{9, 10, ∞9 } {2, 9, ∞2 } {2, 12, 14} {0, 5, 8} {5, 17, ∞8 } {10, 11, ∞6 } {2, ∞5 , ∞8 } {10, 15, ∞1 } {10, ∞3 , ∞5 } {3, 8, ∞6 } {0, ∞4 , ∞5 } {11, 15, ∞8 } {1, 3, 14} {7, 8, ∞7 } {7, 10, 17} {7, 13, 14} {1, 12, 13} {12, ∞6 , ∞8 } {13, 16, ∞10 } {8, 10, ∞10 } {1, ∞4 , ∞9 } {4, ∞1 , ∞9 } {13, ∞2 , ∞8 } {3, 6, 7}
{15, ∞2 , ∞7 } {5, 10, ∞2 } {5, 12, 16} {16, ∞6 , ∞7 } {0, 6, ∞7 } {9, 12, ∞1 } {0, 11, 13} {6, 9, 17} {0, 16, ∞8 } {0, ∞2 , ∞9 } {6, 16, ∞9 } {3, 13, ∞5 } {2, 10, ∞7 } {1, 5, 6} {2, ∞6 , ∞9 } {1, 8, ∞1 } {7, ∞1 , ∞8 } {7, ∞3 , ∞6 } {8, 9, ∞8 } {4, 14, 15} {1, ∞6 , ∞10 } {2, 6, ∞4 } {2, 4, 17} {5, 11, ∞3 }
{2, 8, 13} {17, ∞2 , ∞10 } {9, 16, ∞3 } {5, 15, ∞10 } {5, ∞1 , ∞5 } {5, ∞4 , ∞7 } {6, 8, 11} {10, 16, ∞4 } {6, 12, ∞2 } {11, 12, ∞10 } {3, 10, ∞8 } {11, 16, ∞5 } {6, ∞5 , ∞10 } {7, 9, ∞5 } {7, 11, ∞2 } {12, 17, ∞3 } {4, 8, ∞3 } {13, 15, 17} {1, 16, ∞2 } {13, ∞3 , ∞9 } {14, 16, 17} {4, ∞2 , ∞5 } {4, ∞4 , ∞8 } {4, 5, 9}
Let C1 be the set consisting of the following base triples:
{0, 8, 12} {∞7 , 0, 5}
{1, 5, 7} {∞8 , 1, 6}
{∞4 , 1, 2} {∞9 , 0, 7}
{∞5 , 0, 3} {∞10 , 1, 8}
{∞6 , 1, 4}
Let C2 be the set consisting of the following base triples:
{0, 8, 14} {∞7 , 0, 3}
{1, 3, 7} {∞8 , 1, 4} ⋃
{∞4 , 0, 7} {∞9 , 0, 5}
{∞5 , 1, 2} {∞10 , 1, 6}
{∞6 , 1, 8}
The 27 unordered pairs 0≤i≤8 {{2i, 2i + 1}, {2i, 2i + 2}, {2i + 1, 2i + 9}} can be partitioned into three matching Pj on Z18 (j = 1, 2, 3), where each Pj is listed as follows: P1 = i∈{0,6,12} {{i, i + 2}, {i + 1, i + 9}, {i + 4, i + 5}}, P2 = {{x + 2, y + 2} : {x, y} ∈ P1 } and P3 = {{x + 4, y + 4} : {x, y} ∈ P1 }.
⋃
For each i ∈ Z18 , let Ai = {{x + i, y + i, z + i} : {x, y, z } ∈ A0 }.
Define dev (Cj ) = {{x + 2a, y + 2a, z + 2a} : {x, y, z } ∈ Cj }, dev (Cj+2 ) = {{x + 1, y + 1, z + 1} : {x, y, z } ∈ dev (Cj )} for j = 1, 2. Define Bj , j = 1, 2, 3, 4, as follows:
(
) 3 ⋃ B1 = dev (C1 ) ∪ D1 ∪ {{∞k , x, y} : {x, y} ∈ Pk } ; k=1
(
) 2 ⋃ B2 = dev (C2 ) ∪ D2 ∪ {{∞k , x, y} : {x, y} ∈ Pk+1 } ∪ ({{∞3 , x, y} : {x, y} ∈ P1 }); k=1
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( B3 = dev (C3 ) ∪ D3 ∪
)
–
5
)
3 ⋃ {{∞k , x + 1, y + 1} : {x, y} ∈ Pk } ; k=1
(
) 2 ⋃ B4 = dev (C4 ) ∪ D4 ∪ {{∞k , x + 1, y + 1} : {x, y} ∈ Pk+1 } ∪ {{∞3 , x + 1, y + 1} : {x, y} ∈ P1 }. k=1
It is readily checked that each of (X , G , Ai ) (i ∈ Z18 ) and (X , G , Bj ) (j = 1, 2, 3, 4) is a GDD(2, 3, 28) of type 212 41 . Furthermore, we can check that these 22 block sets of GDDs are pairwise disjoint. Therefore, an LS(212 41 ) exists. Obviously, this LS(212 41 ) contains an LS(23 41 ) (Y , H, {Dj : j = 1, 2, 3, 4}). □ Lemma 3.2 ([5]). Suppose there exists an LS(2n 41 ), where n ̸ = 6. Then there exists an LS(23n 41 ). Lemma 3.3. There is an LS(2n 41 ) for n ∈ {30, 36, 48, 144}. Proof. For n = 36, it exists by Lemma 3.2 since there is an LS(212 41 ) by Lemma 3.1. For n = 30, start with an HF ((33 : 2), 2, 4, 3), which exists in [8, Design 1.B]. Applying Lemma 2.4 gives a PHF ((93 : 5), 2). Since there is an LS(212 41 ) containing an LS(23 41 ) by Lemma 3.1, applying Lemma 2.1 yields an LS(2n 41 ). For n = 48, from the proof of [13, Lemma 4.2], we can obtain an HF ((35 : 2), 2, 4, 3) by deleting the blocks generated by {0, i, 2i, 15 + i}, i ∈ {1, 2, 3, 4, 6, 7} and adjoining four new points to each of four subdesigns GDD(2, 3, 30)s of type 65 , respectively. For completeness, we also list the base blocks which generate the required blocks of an HF ((35 : 2), 2, 4, 3) on Z30 ∪ {x, y, z , w} with groups Gi = {i, i + 15}, 0 ≤ i < 15, G15 = {x, y} and G16 = {z , w} and holes Hj = {Gj , Gj+5 , Gj+10 , G15 , G16 }, 0 ≤ j < 5 and H5 = {G15 , G16 }, under the action of the automorphism (0 1 2 · · · 29)(x y)(z w).
{x, 0, 1, 2} {x, 1, 7, 24} {z , 1, 5, 23} {0, 1, 5, 24} {0, 1, 25, 28} {0, 3, 10, 13}
{x, 1, 3, 15} {z , 0, 1, 8} {z , 0, 7, 16} {0, 1, 6, 12} {0, 2, 4, 9} {0, 3, 12, 22}
{x, 0, 3, 11} {z , 1, 2, 20} {0, 1, 4, 18} {0, 1, 10, 23} {0, 2, 5, 16} {0, 4, 10, 17}
{x, 1, 4, 22} {z , 0, 2, 26} {1, 2, 8, 10} {0, 1, 11, 13} {0, 2, 6, 10} {0, 4, 12, 20}
{x, 0, 4, 21} {z , 1, 3, 14} {1, 3, 9, 22} {0, 1, 14, 20} {0, 2, 11, 22} {0, 5, 12, 19}
{x, 1, 5, 12} {z , 0, 3, 9} {0, 3, 7, 19} {0, 1, 17, 21} {0, 2, 12, 18} {0, 5, 13, 18}
{x, 0, 6, 22} {z , 1, 4, 17} {0, 1, 3, 26} {0, 1, 22, 27} {0, 3, 8, 24} {0, 5, 14, 23}
Applying Lemma 2.4 gives a PHF ((95 : 5), 2). Since there is an LS(212 41 ) containing an LS(23 41 ) by Lemma 3.1, applying Lemma 2.1 yields an LS(2n 41 ). For n = 144, it can be obtained by Lemma 3.2 with the known LS(248 41 ). □ By Lemmas 3.1, 3.3 and Theorem 1.2, the existence of an LS(2n 41 ) is completely determined, i.e., Theorem 1.3 holds. From the definition of an LS(2n 41 ) and simple computation, the number of triples from distinct groups that do not occur 4n(n−1) 4n(n−1) . It is easy to see that these triples cover every pair of points from two in any GDD(2, 3, 2n + 4) of type 2n 41 is 3 3
distinct groups of size 2 twice, i.e., these triples form the block set of a GDD(2, 3, 2n) of type 2n with index 2. It is 3 4n(n−1) interesting if these triples can be partitioned into two parts, each being the block set of a GDD(2, 3, 2n) of type 2n . If 3 n 1 they do, then this LS(2 4 ) is denoted by LS + (2n 41 ). For example, for n = 3, let {F0 , F1 , F2 , F3 } be a one-factorization of complete tripartite graph on Z3 ×Z2 with three partite sets {i}×Z2 , i ∈ {0, 1, 2}. Denote S = {∞0 , ∞1 , ∞2 , ∞3 }. Then, for i ∈ Z4 , each {{∞i+j , a, b} : {a, b} ∈ Fj , j ∈ Z4 } is the block set of GDD(2, 3, 10) of type 23 41 , they form an LS(23 41 ) on (Z3 ×Z2 ) ∪ S with group set {{k}×Z2 : k ∈ Z3 } ∪ {S }, and each {{(0, a0 ), (1, a1 ), (2, a2 )} : a0 , a1 , a2 ∈ Z2 , a0 + a1 + a2 ≡ l (mod 2)}, l ∈ Z2 , is the block set of a GDD(2, 3, 6) of type 23 with group set {{k}×Z2 : k ∈ Z3 }. So, there is an LS + (23 41 ). From the construction of an LS(2mn+s−2 41 ) from a PHF ((mn : s), 2), one can obtain an LS + (2mn+s−2 41 ) if there is an LS + (2m+s−2 41 : 2s−2 41 ). We end this paper with the following conjecture. 4n(n−1)
Conjecture. There is an LS + (2n 41 ) for any n ≡ 0 (mod 3). Acknowledgments The authors would like to thank the reviewers for many comments and Prof. L. Zhu for helpful suggestions on this topic. References [1] H. Cao, L. Ji, L. Zhu, Large sets of disjoint packings on 6k + 5 points, J. Combin. Theory (A) 108 (2004) 169–183. [2] H. Cao, J. Lei, L. Zhu, Large sets of disjoint group-divisible designs with block size three and type 2n 41 , J. Combin. Des. 9 (2001) 285–296. [3] H. Cao, J. Lei, L. Zhu, Further results on large sets of disjoint group-divisible designs with block size three and type 2n 41 , J. Combin. Des. 11 (2003) 24–35. [4] H. Cao, J. Lei, L. Zhu, Constructions of large sets of disjoint group-divisible designs LS(2n 41 ) using a generalization of *LS(2n ), Discrete Math. 338 (2015) 1449–1459.
Please cite this article in press as: Y. Chang, et al., A completion of LS(2n 41 ), Discrete Mathematics (2016), http://dx.doi.org/10.1016/j.disc.2016.10.020
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