Uniformly resolvable designs with block sizes 3 and 4

Discrete Mathematics 339 (2016) 1069–1085

Contents lists available at ScienceDirect

Discrete Mathematics journal homepage: www.elsevier.com/locate/disc

Uniformly resolvable designs with block sizes 3 and 4 Hengjia Wei a , Gennian Ge a,b,∗ a

School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

b

Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100048, China

article

info

Article history: Received 31 July 2015 Received in revised form 29 October 2015 Accepted 30 October 2015

Keywords: Uniformly resolvable design Resolvable group divisible design Frame

abstract A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k. Such a class is denoted k-pc and for a given k the number of k-pcs is denoted rk . Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0 (mod 12). It has been shown that almost all URDs with permissible r3 and r4 exist for v ≡ 0 (mod 24), v ≡ 0 (mod 60), v ≡ 36 (mod 144) or v ≡ 36 (mod 108). In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v = 12, r3 = 1 and r4 = 3. © 2015 Published by Elsevier B.V.

1. Introduction Let v and λ be positive integers, and let K and M be two sets of positive integers. A group divisible design, denoted GDD(K , M ; v), is a triple (X , G, B ) where X is a set of points, G is a partition of X into groups, and B is a collection of subsets of X , called blocks, such that 1. 2. 3. 4.

|B| ∈ K for each B ∈ B , |G| ∈ M for each G ∈ G, |B ∩ G| ≤ 1 for each B ∈ B and each G ∈ G, and each pair of elements of X from distinct groups is contained in exactly one block.

If K = {k}, respectively M = {m}, then the GDD(K , M ; v) is simply denoted GDD(k, M ; v), respectively GDD(K , m; v). A GDD(K , 1; v) is called a pairwise balanced design and denoted PBD(K ; v). A GDD(k, m; mk) is called a transversal design and u u denoted TD(k, m). We usually use an ‘‘exponential’’ notation to describe the multiset M: a K -GDD of type g1 1 g2 2 . . . gsus is a GDD in which every block has size from the set K and in which there are ui groups of size gi , i = 1, 2, . . . , s. In a GDD(K , M ; v)(X , G, B ) a parallel class is a set of blocks, which partitions X . If B can be partitioned into parallel classes, then the GDD(K , M ; v) is said to be resolvable and denoted RGDD(K , M ; v). Analogously, a resolvable PBD(K ; v) is denoted RPBD(K ; v). A parallel class is said to be uniform if it contains blocks of only one size k (k-pc). If all parallel classes of an RPBD(K ; v) are uniform, the design is said to be uniformly resolvable. Here, a uniformly resolvable design RPBD(K ; v) is denoted URD(K ; v). In a URD(K ; v) the number of parallel classes with blocks of size k is denoted rk , k ∈ K . In [19], Rees introduced the notation of URDs and showed that all admissible URDs({2, 3}; v) exist. For K = {2, 4}, almost all URD(K ; v) have been constructed in [8,25], with a small number of cases unsettled. For K = {3, 4}, we summarize the known results as follows.

∗

Corresponding author at: School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China. E-mail address: [email protected] (G. Ge).

http://dx.doi.org/10.1016/j.disc.2015.10.042 0012-365X/© 2015 Published by Elsevier B.V.

1070

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

Theorem 1.1. The necessary conditions for the existence of a URD({3, 4}; v) with r3 , r4 > 0 are v ≡ 0 (mod 12), r4 is odd, v−1−3r4 1 ≤ r4 ≤ v3 − 1 and r3 = . 2 Theorem 1.2 ([7,23–25]). Let v ≡ 0 (mod 12). There exists a URD({3, 4}; v) with r4 = 1, 3, 5, 7 or 9, except for v = 12 and r4 ∈ {3, 5, 7, 9} or v = 24 and r4 = 9. Theorem 1.3 ([24,25]). Let v ≡ 0 (mod 12). There exists a URD({3, 4}; v) with r3 = 1, 4, 7 or 10, except for v = 12 and r3 ∈ {1, 7, 10}, and possibly excepting: 1. r3 = 7 and v ∈ {84, 108, 132, 156, 204, 228, 276, 348, 372, 444}; or 2. r3 = 10 and v ∈ {108, 132, 156, 204, 228, 276, 348, 372, 492}. Theorem 1.4 ([24]). There exist all admissible URDs({3, 4}; v), v ≡ 0 (mod 12), v < 200, except when v = 12 and r4 = 3 and possibly excepting: 1. 2. 3. 4. 5.

v v v v v

= 84: r4 = 23; = 108: r4 ∈ {29, 31}; = 120: r4 ∈ {27, 29, 31}; = 132: r4 ∈ {35, 37, 39}; = 156: r4 ∈ {41, 43, 45, 47}.

Theorem 1.5 ([25,24]). There exist all admissible URDs({3, 4}; v) for v ≡ 0 (mod 24), possibly excepting: 1. 2. 3. 4. 5. 6. 7. 8.

v v v v v v v v

= 120 and r4 ∈ {(v/3) − 13, (v/3) − 11, (v/3) − 9}; = 264 and r4 = (v/3) − 9; = 408 and r4 ∈ {(v/3) − 15, (v/3) − 13, (v/3) − 11, (v/3) − 9}; = 456 and r4 ∈ {(v/3) − 11, (v/3) − 9}; = 552 and r4 ∈ {(v/3) − 13, (v/3) − 11, (v/3) − 9}; = 984 and r4 ∈ {(v/3) − 13, (v/3) − 11, (v/3) − 9}; = 1128 and r4 = (v/3) − 9; = 3288 and r4 = (v/3) − 9.

Theorem 1.6 ([24]). There exist all admissible URDs({3, 4}; v) for v ≡ 0 (mod 60). In this paper, we consider the entire existence problem of the URD({3, 4}; v) and show that the necessary conditions are also sufficient, with only one exception. Theorem 1.7. There exists a URD({3, 4}; v) with r3 , r4 > 0 if and only if v ≡ 0 (mod 12), r4 is odd and 1 ≤ r4 ≤ v3 − 1, except for v = 12 and r4 = 3. 2. Preliminaries A group divisible design (X , G, B ) is called frame resolvable (and is referred to as a frame) if its block set B admits a partition into holey parallel classes, each holey parallel class being a partition of X \H for some group H ∈ G. The groups in a frame are often referred to as holes. The hole type of a frame is just its group type as a GDD. It is well known that in a k-frame, each hole must have size a multiple of k − 1; in fact the number of holey parallel classes with respect to a given hole H is precisely |H |/(k − 1). Theorem 2.1 ([5,6,10,17,13,18,22,27]). The necessary conditions for the existence of a k-frame of type hu , namely, u ≥ k + 1, h ≡ 0 (mod k − 1) and h(u − 1) ≡ 0 (mod k), are also sufficient for k = 2; k = 3; and for k = 4, and possibly excepting: 1. h = 36 and u = 12; 2. h ≡ 6 (mod 12) and (a) h = 6 and u ∈ {7, 23, 27, 35, 39, 47}; (b) h = 18 and u ∈ {15, 23, 27}; (c) h ∈ {30, 66, 78, 114, 150, 174, 222, 246, 258, 282, 318, 330, 354, 534} and u ∈ {7, 23, 27, 39, 47}; (d) h ∈ {n : 42 ≤ n ≤ 11238}\{66, 78, 114, 150, 174, 222, 246, 258, 282, 318, 330, 354, 534} and u ∈ {23, 27}.

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1071

A K -frame is called uniform if each holey parallel class is of only one block size. It is called completely uniform if for each hole G the parallel classes which partitions X \G are all of one block size. In this paper, we use mostly K = {3, 4}. A {3, 4}frame of type (g1 ; 3a1 4b1 )u1 (g2 ; 3a2 4b2 )u2 . . . (gs ; 3as 4bs )us has ui groups of size gi , and each group of size gi has ai holey pcs of block size 3 and bi holey pcs of block size 4. An incomplete group divisible design (IGDD) is a quadruple (X , H , G, B ) where X is a set of points, H is a subset of X (called the hole), G is a partition of X into groups, and B is a collection of subsets of X (blocks) such that (i) for each block B ∈ B , |B ∩ H | ≤ 1, and (ii) any pair of points from X which are not both in H occurs either in some group or in exactly one block, but not both. A K -IGDD of type (g1 , h1 )u1 (g2 , h2 )u2 . . . (gs , hs )us is an IGDD in which every block has size from the set K and in which there are ui groups of size gi , each of which intersects the hole in hi points, i = 1, 2, . . . , s. A K -IGDD (X , H , G, B ) is said to be uniformly resolvable and denoted by K -IUGDD if its blocks can be partitioned into uniform parallel classes and holey uniform parallel classes, the latter partitioning X \H. The numbers of uniform parallel classes, holey uniform parallel classes with blocks of size k are denoted rk and rk0 , respectively. If |G| = 1 for G ∈ G, then the IUGDD is called an incomplete uniformly resolvable design and denoted IURD(K ; v, h) where |X | = v and |H | = h. The following construction can be found in [25]. Construction 2.2 (Filling in Groups). Suppose there exist a k1 -RGDD of type hu and a URD({k1 , k2 }; h) with rk2 = i, then there exists a URD({k1 , k2 }; hu) with rk2 = i and an IURD({k1 , k2 }; hu, h) with rk1 =

rk2 = 0 k2 -pcs and

rk02

h(u−1) k 1 −1

k1 -pcs, rk01 =

h−1−(k2 −1)i k1 −1

holey k1 -pcs,

= i holey k2 -pcs.

The following construction is a generalization of [25, Construction 3.2]. The proof is similar and we omit here. a

b

Construction 2.3 (Frame Construction). Suppose there exists a uniform {k1 , k2 }-frame of type {(gi ; k1i k2i ) : i = 1, 2, . . . , s}. s 0 Let v = i=1 gi . If, for each i = 1, 2, . . . , s − 1, there exists an IURD({k1 , k2 }; gi + h, h) with rk1 = ai , rk1 = a, rk2 = bi , and

rk02 = b (therefore (k1 − 1)a + (k2 − 1)b = h − 1) and if there exists a URD({k1 , k2 }; gs + h) with rk1 = as + a and rk2 = bs + b, then a URD({k1 , k2 }; v + h) with rk2 =

s

i =1

bi + b exists.

Construction 2.4 (Weighting). Let (X , G, B ) be a GDD, and let w, a, b : X → Z≥0 be functions with w(x) = (k1 −1)a(x)+(k2 − a(x) b(x)

1)b(x) for all x ∈ X . Suppose that for each block B ∈ B , there exists a uniform {k1 , k2 }-frame of type {(w(x); k1 k2 ) : x ∈ B}.   Then there is a uniform {k1 , k2 }-frame of type {(



x∈G

w(x); k1

x∈G a(x)

k2

x∈G b(x)

) : G ∈ G}.

Proof. Let x ∈ G ∈ G be an arbitrary fixed point of the GDD. For any other point y ∈ X not in the same group G ∈ G with x, there exists precisely one block B ∈ B such that B contains both x and y. It is easy to see that ∪x∈B∈B (B\{x}) forms a partition of X \G, where x ∈ G. The remaining verification is then straightforward.  The following results provide us ingredient designs when we apply Constructions 2.3 and 2.4. Theorem 2.5 ([5,9,11,12,15,17,14,20,22,25–27]). The necessary conditions for the existence of a k-RGDD of type hu , namely, u ≥ k, hu ≡ 0 (mod k) and h(u − 1) ≡ 0 (mod k − 1), are also sufficient for k = 2; k = 3, except for (h, u) ∈ {(2, 3), (2, 6), (6, 3)}; and for k = 4, except for (h, u) ∈ {(2, 4), (2, 10), (3, 4), (6, 4)} and possibly excepting: 1. h h 2. h 3. h

≡ 2, 10 (mod 12): h = 2 and u ∈ {46, 70, 82, 94, 100, 118, 130, 178, 202, 214, 250, 334}; h = 10 and u ∈ {4, 94}; = 26 and u ∈ {10, 70, 82}; h ∈ {38, 58, 74, 82, 86, 94, 106} and u = 10. ≡ 6 (mod 12): h = 6 and u ∈ {6, 68}; h = 18 and u ∈ {38, 62}. ≡ 0 (mod 12): h = 36 and u ∈ {14, 15, 18, 23}.

Theorem 2.6 ([2,16,17,27,28]). The necessary conditions for the existence of a 5-GDD of type g u , namely, u ≥ 5, g (u − 1) ≡ 0 (mod 4) and g 2 u(u − 1) ≡ 0 (mod 20), are also sufficient, except when (g , u) ∈ {(2, 5), (2, 11), (3, 5), (6, 5)}, and possibly where 1. g = 3 and u ∈ {45, 65}; 2. g ≡ 2, 6, 14, 18 (mod 20) and (a) g = 2 and u ∈ {15, 35, 71, 75, 95, 111, 115, 195, 215}; (b) g = 6 and u ∈ {15, 35, 75, 95}; (c) g ∈ {14, 18, 22, 26} and u ∈ {11, 15, 71, 111, 115}; (d) g ∈ {34, 46, 62} and u ∈ {11, 15}; (e) g ∈ {38, 58} and u ∈ {11, 15, 71, 111}; (f) g = 2α with gcd(30, α) = 1 and 33 ≤ α ≤ 2443, and u = 15;

1072

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

3. g ≡ 10 (mod 20) and (a) g = 10 and u ∈ {5, 7, 15, 23, 27, 33, 35, 39, 47}; (b) g = 30 and u = 15; (c) g = 50 and u ∈ {15, 23, 27}; (d) g = 90 and u = 23; (e) g = 10α with α ∈ {7, 11, 13, 17, 35, 55, 77, 85, 91, 119, 143, 187, 221}, and u = 23. Theorem 2.7 ([3,4,16,21,27]). A 5-GDD of type g 5 m1 exists whenever g ≡ 0 (mod 4), m ≡ 0 (mod 4) and m ≤ 4g /3, with the possible exceptions of (g , m) ∈ {(12, 4), (12, 8)}. Theorem 2.8 ([1,5]). Let m be a positive integer. Then: (i) (ii) (iii) (iv)

a TD(4, m) exists if m ̸∈ a TD(5, m) exists if m ̸∈ a TD(6, m) exists if m ̸∈ a TD(m + 1, m) exists if

{2, 6}; {2, 3, 6, 10}; {2, 3, 4, 6, 10, 22}; m is a prime power.

A double group divisible design (DGDD) is a quadruple (X , H , G, B ) where X is a set of points, H and G are partitions of X (into holes and groups, respectively) and B is a collection of subsets of X (blocks) such that (i) for each block B ∈ B and each hole H ∈ H , |B ∩ H | ≤ 1, and (ii) any pair of distinct points from X which are not in the same hole occurs either in some group or in exactly one block, but not both. A K -DGDD of type (g1 , hv1 )u1 (g2 , hv2 )u2 . . . (gs , hvs )us is a double group divisible design in which every block has size from the set K and in which there are ui groups of size gi , each of which intersects each of the v holes in hi points. (Thus, gi = hi v for i = 1, 2, . . . , s. Not every DGDD can be expressed this way, of course, but this is the most general type that we will require.) Thus, for example, a modified group divisible design K -MGDD of type g u is a K -DGDD of type (g , 1g )u . A K -DGDD (X , H , G, B ) is called frame resolvable if its block set B admits a partition into holey parallel classes, each of which partitions X \G for some G ∈ G. Furthermore, if each holey parallel class is of only one block size it is called uniform. a b a b a b A uniformly frame resolvable {k1 , k2 }-DGDD of type (g1 , hv1 ; k11 k21 )u1 (g2 , hv2 ; k12 k22 )u2 . . . (gs , hvs ; k1s k2s )us has ui groups of size gi , and each group of size gi has ai holey k1 -pcs and bi holey k2 -pcs. Construction 2.9 (Filling in Holes). Suppose that there is a uniformly frame resolvable {k1 , k2 }-DGDD of type

(g , hv ; k11 k21 )u1 (g , hv ; k12 k22 )u2 . . . (g , hv ; ka1s kb2s )us s and that there is a k1 -frame of type hu with u = i=1 ui . Then there is a uniform {k1 , k2 }-frame of type a

b

a1 + k h−1 1

(g ; k 1

a

b

b

a2 + k h−1 1

k21 )u1 (g ; k1

b

as + k h−1 1

k22 )u2 . . . (g ; k1

k2s )us . b

Proof. Fill all holes of the {k1 , k2 }-DGDD with the k1 -frame to obtain the uniform {k1 , k2 }-frame, noting that each group of the k1 -frame has k h−1 holey k1 -pcs.  1

3. Admissible URD({3, 4}; v) for small v Lemma 3.1. There exists a URD({3, 4}; 84) with r4 = 23. Proof. We start with a {3, 4}-URGDD of type 127 with r4 = 22 constructed below and fill in each group with a URD({3, 4}; 12) to obtain the desired design. The requisite URGDD is constructed on X = Z28 × {0, 1, 2} with groups {{i, i + 7, i + 14, i + 21} × {0, 1, 2}} for i = 0, 1, 2, . . . , 6. Develop the following base blocks (+2 (mod 28), −) and (−, +1 (mod 3)). Note that each of the base blocks marked ∗ generates two 4-pcs, all the blocks marked a generate fourteen 4-pcs, and the last two blocks generate three 3-pcs.

{(5, 0), (11, 2), (6, 2), (8, 2)}∗ {(14, 0), (13, 2), (8, 0), (11, 2)}∗ {(8, 0), (16, 0), (24, 2), (25, 2)}a {(2, 0), (26, 1), (21, 1), (18, 0)}a {(11, 0), (13, 2), (3, 1), (19, 0)}a {(5, 0), (9, 0), (20, 0), (22, 2)}a {(24, 1), (7, 0), (8, 0)}

{(20, 0), (1, 1), (10, 2), (11, 0)}∗ {(11, 0), (26, 1), (13, 1), (16, 1)}∗ {(10, 0), (4, 2), (0, 1), (6, 0)}a {(1, 0), (17, 0), (12, 2), (7, 0)}a {(15, 0), (23, 1), (14, 1), (27, 2)}a {(0, 2), (11, 2), (15, 1)}. 

Lemma 3.2. There exists a URD({3, 4}; 108) with r4 ∈ {29, 31}.

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1073

Proof. We start with a {3, 4}-URGDD of type 129 with r4 = 28 or 30 and fill in the groups to obtain the desired design. The requisite URGDDs are constructed on X = Z36 ×{0, 1, 2} with groups {{i, i + 9, i + 18, i + 27}×{0, 1, 2}} for i = 0, 1, 2, . . . , 8. Develop the following base blocks (+2 (mod 36), −) and (−, +1 (mod 3)). For r4 = 28:

{(15, 0), (34, 2), (8, 1), (13, 0)}∗ {(31, 0), (18, 2), (5, 0), (12, 0)}∗ {(7, 0), (0, 0), (1, 1), (22, 2)}∗ {(0, 0), (5, 1), (8, 2), (3, 0)}a {(18, 0), (19, 0), (20, 0), (24, 1)}a {(11, 0), (7, 1), (15, 0), (35, 1)}a {(12, 0), (13, 2), (29, 0), (17, 0)}a {(7, 2), (24, 1), (35, 1)} {(16, 0), (33, 2), (0, 1)}

{(21, 0), (32, 2), (35, 1), (6, 0)}∗ {(33, 0), (27, 2), (8, 0), (2, 0)}∗ {(1, 0), (30, 2), (27, 2), (16, 0)}a {(22, 0), (14, 2), (26, 2), (34, 2)}a {(31, 0), (25, 0), (9, 0), (32, 1)}a {(10, 0), (21, 1), (23, 0), (33, 2)}a {(6, 0), (4, 1), (28, 0), (2, 0)}a {(14, 0), (1, 0), (34, 2)} {(0, 2), (21, 1), (35, 0)}.

Note that each of the base blocks marked ∗ generates two 4-pcs, the blocks marked a generate eighteen 4-pcs, and the two blocks of size 3 on each row generate three 3-pcs. For r4 = 30:

{(23, 0), (17, 1), (30, 1), (24, 1)}∗ {(4, 0), (29, 0), (18, 2), (35, 1)}∗ {(19, 0), (23, 0), (29, 0), (31, 2)}a {(6, 0), (32, 2), (34, 0), (22, 2)}a {(30, 0), (18, 0), (10, 1), (15, 2)}a {(4, 0), (28, 1), (11, 2), (8, 1)}a

{(7, 0), (29, 0), (6, 2), (28, 1)}∗ {(13, 0), (35, 2), (27, 2), (2, 1)}a {(21, 0), (14, 2), (20, 0), (16, 1)}a {(26, 0), (1, 0), (24, 0), (3, 1)}a {(25, 0), (12, 1), (33, 1), (9, 1)}a {(5, 0), (0, 0), (7, 0), (17, 1)}a

{(4, 0), (14, 2), (10, 2), (33, 1)}b

{(18, 0), (32, 0), (13, 0), (17, 1)}b

{(7, 0), (24, 0), (23, 0), (27, 2)}b {(2, 2), (5, 2), (10, 0)}

{(0, 1), (15, 1), (25, 0)}.

Note that each of the base blocks marked ∗ generates two 4-pcs, the blocks marked a generate eighteen 4-pcs, the blocks marked b generate six 4-pcs, and the last two blocks generate three 3-pcs.  Lemma 3.3. There exists a URD({3, 4}; 120) for each r4 ∈ {27, 29, 31}. Proof. We start with a {3, 4}-URGDD of type 1210 with r4 = 26, 28 or 30 and fill in the groups to obtain the desired design. The requisite URGDDs are constructed on X = Z40 × {0, 1, 2} with groups {{i, i + 10, i + 20, i + 30} × {0, 1, 2}} for i = 0, 1, 2, . . . , 9. Develop the following base blocks (+2 (mod 40), −) and (−, +1 (mod 3)), noting that each of the base blocks marked ∗ generates two 4-pcs and the blocks marked a generate twenty 4-pcs; the two blocks of size 3 on each row generate three 3-pcs. For r4 = 26:

{(37, 0), (8, 0), (19, 2), (10, 1)}∗ {(35, 0), (32, 0), (9, 0), (14, 0)}∗ {(4, 0), (26, 1), (32, 1), (28, 1)}a {(22, 0), (3, 1), (1, 1), (9, 1)}a {(7, 0), (2, 1), (5, 2), (10, 1)}a {(37, 0), (0, 2), (13, 2), (31, 1)}a {(11, 0), (23, 2), (35, 0), (27, 1)}a {(33, 1), (12, 2), (0, 1)} {(22, 1), (9, 1), (13, 2)} {(20, 0), (3, 2), (6, 2)} {(23, 1), (14, 1), (9, 0)} {(9, 0), (2, 2), (35, 1)}

{(39, 0), (32, 1), (38, 2), (37, 1)}∗ {(30, 0), (24, 1), (16, 0), (15, 1)}a {(6, 0), (8, 2), (20, 2), (19, 2)}a {(12, 0), (36, 0), (29, 1), (25, 1)}a {(34, 0), (17, 1), (38, 2), (39, 1)}a {(33, 0), (14, 0), (21, 0), (18, 1)}a {(15, 0), (9, 2), (0, 0)} {(32, 2), (24, 0), (7, 0)} {(3, 1), (18, 1), (35, 0)} {(0, 2), (16, 0), (1, 2)} {(0, 1), (11, 2), (22, 0)}.

1074

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

For r4 = 28:

{(8, 0), (19, 2), (37, 1), (26, 0)}∗ {(26, 0), (7, 2), (8, 1), (25, 2)}∗ {(3, 0), (37, 0), (30, 2), (15, 1)}a {(21, 0), (17, 1), (12, 0), (38, 1)}a {(23, 0), (25, 2), (9, 0), (6, 0)}a {(39, 0), (35, 2), (11, 2), (7, 2)}a {(5, 0), (31, 1), (10, 0), (22, 2)}a {(36, 0), (39, 1), (23, 0)} {(27, 1), (6, 1), (33, 2)} {(15, 0), (0, 2), (29, 1)} {(23, 0), (15, 2), (32, 2)}

{(7, 0), (1, 1), (6, 2), (0, 1)}∗ {(36, 0), (29, 0), (22, 0), (31, 1)}∗ {(2, 0), (1, 0), (33, 0), (4, 0)}a {(27, 0), (29, 0), (14, 1), (26, 1)}a {(0, 0), (8, 1), (16, 1), (13, 0)}a {(34, 0), (28, 1), (19, 0), (32, 1)}a {(18, 0), (24, 0), (20, 1), (36, 1)}a {(8, 2), (19, 2), (34, 1)} {(32, 2), (24, 0), (29, 0)} {(12, 1), (5, 2), (28, 0)} {(0, 0), (4, 1), (19, 1)}.

For r4 = 30:

{(2, 0), (8, 1), (17, 0), (35, 0)}∗ {(11, 0), (25, 1), (4, 1), (38, 1)}∗ {(3, 0), (38, 1), (24, 0), (9, 1)}∗ {(16, 0), (23, 0), (39, 1), (27, 1)}a {(12, 0), (20, 0), (24, 0), (26, 2)}a {(22, 0), (34, 2), (29, 1), (38, 0)}a {(8, 0), (36, 2), (0, 1), (37, 0)}a {(32, 0), (30, 0), (35, 0), (19, 1)}a {(22, 0), (21, 0), (39, 1)} {(31, 1), (3, 0), (34, 0)} {(22, 0), (4, 2), (21, 2)}

{(0, 0), (11, 2), (17, 2), (26, 0)}∗ {(21, 0), (18, 1), (19, 0), (0, 1)}∗ {(31, 0), (5, 0), (33, 2), (9, 2)}a {(10, 0), (1, 0), (3, 1), (28, 2)}a {(11, 0), (14, 0), (6, 2), (15, 0)}a {(17, 0), (4, 0), (25, 1), (2, 2)}a {(13, 0), (18, 2), (7, 1), (21, 0)}a {(4, 2), (20, 1), (39, 2)} {(0, 1), (16, 2), (39, 2)} {(18, 1), (3, 1), (11, 0)}. 

Lemma 3.4. There exists a URD({3, 4}; 132) for each r4 ∈ {35, 37, 39}. Proof. We start with a {3, 4}-URGDD of type 1211 with r4 = 34, 36 or 38 and fill in the groups to obtain the desired design. The requisite URGDDs are constructed on X = Z44 × {0, 1, 2} with groups {{i, i + 11, i + 22, i + 33} × {0, 1, 2}} for i = 0, 1, 2, . . . , 10. Develop the following base blocks (+2 (mod 44), −) and (−, +1 (mod 3)). Note that each of the blocks marked ∗ generates two 4-pcs and the blocks marked a generate twenty-two 4-pcs; the two blocks of size 3 on each row generate three 3-pcs. For r4 = 34:

{(16, 0), (21, 0), (7, 1), (6, 2)}∗ {(22, 0), (29, 2), (31, 2), (4, 1)}∗ {(22, 0), (17, 1), (19, 0), (4, 0)}∗ {(29, 0), (17, 1), (14, 1), (13, 1)}a {(5, 0), (9, 1), (30, 1), (32, 2)}a {(19, 0), (6, 1), (18, 0), (31, 0)}a {(27, 0), (23, 1), (43, 0), (7, 0)}a {(42, 0), (36, 2), (22, 2), (28, 1)}a {(25, 0), (16, 2), (8, 0), (39, 1)}a {(21, 0), (33, 1), (24, 1)} {(26, 1), (6, 2), (29, 0)} {(10, 2), (26, 1), (17, 0)}

{(29, 0), (19, 0), (26, 2), (0, 0)}∗ {(10, 0), (27, 2), (41, 2), (20, 2)}∗ {(27, 0), (42, 2), (28, 1), (33, 2)}∗ {(40, 0), (0, 0), (12, 0), (2, 0)}a {(35, 0), (15, 2), (33, 2), (41, 0)}a {(24, 0), (20, 1), (1, 1), (4, 0)}a {(38, 0), (21, 2), (34, 2), (26, 2)}a {(11, 0), (10, 2), (37, 2), (3, 1)}a {(8, 0), (10, 2), (3, 2)} {(0, 0), (19, 1), (29, 2)} {(14, 0), (19, 2), (13, 1)}.

For r4 = 36:

{(5, 0), (3, 1), (42, 1), (0, 1)}∗ {(21, 0), (7, 0), (24, 1), (14, 0)}∗ {(1, 0), (0, 2), (10, 2), (39, 2)}∗ {(4, 0), (19, 2), (42, 2), (5, 0)}∗ {(6, 0), (18, 0), (2, 0), (22, 1)}a {(30, 0), (16, 1), (28, 2), (15, 1)}a {(32, 0), (36, 2), (41, 0), (24, 0)}a {(3, 0), (43, 2), (11, 2), (35, 1)}a {(31, 0), (12, 0), (7, 2), (38, 2)}a {(8, 2), (2, 0), (31, 2)} {(34, 2), (29, 0), (10, 1)}

{(11, 0), (21, 0), (34, 2), (8, 1)}∗ {(11, 0), (9, 0), (30, 2), (12, 2)}∗ {(33, 0), (16, 1), (18, 0), (19, 2)}∗ {(17, 0), (33, 2), (34, 0), (42, 2)}a {(10, 0), (40, 2), (20, 2), (26, 2)}a {(37, 0), (27, 2), (9, 0), (29, 0)}a {(25, 0), (8, 2), (21, 0), (0, 1)}a {(13, 0), (4, 2), (1, 2), (19, 0)}a {(5, 0), (14, 1), (23, 0), (39, 1)}a {(29, 0), (23, 1), (36, 1)} {(0, 0), (23, 2), (27, 1)}.

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1075

For r4 = 38:

{(27, 0), (18, 1), (36, 1), (9, 2)}∗ {(42, 0), (17, 2), (7, 0), (32, 0)}∗ {(26, 0), (7, 2), (1, 1), (0, 1)}∗ {(1, 0), (11, 0), (40, 2), (42, 2)}∗ {(2, 0), (34, 0), (38, 0), (22, 2)}a {(43, 0), (3, 2), (27, 2), (19, 1)}a {(9, 0), (30, 0), (35, 0), (26, 2)}a {(39, 0), (18, 0), (25, 2), (24, 1)}a {(33, 0), (13, 1), (32, 1), (15, 1)}a {(36, 0), (23, 1), (10, 2), (31, 0)}a {(22, 0), (3, 1), (35, 0)}

{(29, 0), (30, 0), (27, 1), (24, 1)}∗ {(29, 0), (42, 0), (35, 2), (12, 0)}∗ {(12, 0), (41, 1), (14, 2), (35, 1)}∗ {(24, 0), (14, 1), (17, 0), (31, 0)}∗ {(37, 0), (29, 0), (42, 1), (41, 0)}a {(1, 0), (6, 2), (16, 0), (4, 1)}a {(5, 0), (17, 2), (21, 0), (7, 1)}a {(14, 0), (8, 0), (0, 1), (28, 1)}a {(11, 0), (20, 0), (12, 2), (40, 0)}a {(0, 2), (4, 1), (41, 2)}. 

Lemma 3.5. There exists a {3, 4}-IUGDD of type (24, 0)5 (36, 36)1 with r4 = 40 and r30 = 6. Proof. The IUGDD is constructed on X = Z120 ∪ S with S = ({a} × Z15 ) ∪ ({b, c , . . . , h} × Z3 ). The groups are {i, i + 5, i + 10, . . . , i + 115}, i = 0, 1, 2, 3, 4, together with S. Develop the following base blocks +1 (mod 120). The elements of form xi ∈ {x} × Zn mean (x, i) and the subscripts are developed modulo the unique subgroup in Z120 of order n. Note that these base blocks and their translates +40 (mod 120) and +80 (mod 120) form a parallel class. Thus we can get 40 parallel classes in total.

{47, 5, 104, 36} {88, 69, 15, a6 } {11, 43, 39, d0 } {84, 97, 35, h0 }

{96, 102, 0, a0 } {67, 58, 31, a3 } {42, 103, 65, e0 }

{20, 101, 68, a12 } {33, 41, 19, b0 } {12, 46, 53, f0 }

{26, 14, 70, a9 } {112, 9, 38, c0 } {90, 37, 74, g0 }

Then develop the blocks {0, 1, 3} and {32, 101, 58} to obtain 6 holey parallel classes.



Lemma 3.6. There exists a URD({3, 4}; 156) for each r4 ∈ {41, 43, 45, 47}. Proof. We start with a {3, 4}-IUGDD of type (24, 0)5 (36, 36)1 from Lemma 3.5. Fill in each group of size 24 with a URD({3, 4}; 24) where (r3 , r4 ) ∈ {(10, 1), (7, 3), (4, 5), (1, 7)} (see Theorem 1.4), and fill in the group of size 36 with a URD({3, 4}; 36) where (r3 , r4 ) ∈ {(16, 1), (13, 3), (10, 5), (7, 7)} (see Theorem 1.4). Then we obtain a URD({3, 4}; 156) with (r3 , r4 ) ∈ {(16, 41), (13, 43), (10, 45), (7, 47)}, as desired.  Combining the above results, Theorem 1.4 can be updated as follows. Theorem 3.7. There exist all admissible URDs({3, 4}; v) for every v ≡ 0 (mod 12), v < 200, except when v = 12 and r4 = 3. Lemma 3.8. There exists a {3, 4}-IUGDD of type (24, 0)7 (36, 36)1 with r4 = 56 and r30 = 6. Proof. The IUGDD is constructed on X = Z168 ∪ S with S = ({a} × Z12 ) ∪ ({b, c , . . . , i} × Z3 ). The groups are {i, i + 7, i + 14, . . . , i + 161}, i = 0, 1, 2, . . . , 6, together with S. Develop the following base blocks +1 (mod 168). Note that these base blocks and their translates +56 (mod 168) and +112 (mod 168) form a parallel class. Thus we can get 56 parallel classes in total.

{71, 34, 66, 95} {67, 153, 145, 93} {72, 78, 13, a3 } {28, 117, 83, e0 } {19, 23, 99, i0 }

{126, 107, 137, 80} {38, 162, 60, a0 } {6, 47, 64, b0 } {92, 102, 52, f0 }

{77, 65, 152, 29} {124, 86, 73, a9 } {87, 160, 56, c0 } {7, 54, 74, g0 }

{59, 26, 98, 44} {55, 161, 2, a6 } {91, 165, 20, d0 } {32, 157, 57, h0 }

Then develop the blocks {0, 1, 3} and {164, 148, 65} to obtain 6 holey parallel classes.



Lemma 3.9. There exists a URD({3, 4}; 204) for each r4 ∈ {11, 13, 15, . . . , 47}. Proof. For r4 = 47, we first construct a {3, 4}-GDD of type 514 on Z204 with groups {i, 4 + i, . . . , 200 + i}, i = 0, 1, 2, 3. Develop the following blocks +2 (mod 204).

{35, 157, 164} {39, 66, 12, 57} {182, 60, 41, 115} {80, 78, 45, 23} {195, 154, 41, 36} {42, 4, 99, 193} {148, 191, 18, 29}

{64, 157, 191} {160, 22, 147, 161} {19, 80, 193, 10} {177, 43, 130, 160} {36, 85, 7, 54} {126, 160, 199, 85} {28, 111, 105, 174}

{16, 95, 6} {201, 115, 80, 94} {32, 141, 142, 131} {53, 186, 15, 160} {17, 116, 2, 15} {196, 173, 199, 54} {0, 7, 65, 158}.

{0, 98, 199} {68, 105, 51, 194} {49, 95, 98, 120} {43, 105, 178, 24} {74, 80, 37, 103} {160, 121, 202, 23}

1076

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

Note that each base block of size 4 generates two parallel classes and each base block of size 3 generates two holey parallel classes. Develop the block {0, 51, 102, 153} + 1 (mod 204) to obtain another parallel class. Thus we have forty-seven 4-pcs in total and for each group there are two holey 3-pcs. Fill in each group with an RPBD({3}; 51). The two holey 3-pcs associated to the same group will consume two 3-pcs of the RPBD({3}; 51) to form parallel classes. Thus we obtain a URD({3, 4}; 204) with r3 = 25 − 2 + 2 × 4 = 31 and r4 = 47, as desired. For the remaining values of r4 , the construction is exactly the same. We list the base blocks for the requisite {3, 4}-GDDs of type 514 in the Appendix.  Lemma 3.10. There exists a URD({3, 4}; 204) for each r4 ∈ {49, 53, 55}. Proof. We start with a uniformly resolvable {3, 4}-DGDD of type (51, 317 )4 with r4 = 32, 36 or 38, which is constructed below. Fill in each group with a 4-frame of type 317 (see Theorem 2.1) and fill in each hole with a URD({3, 4}; 12) to obtain the desired URD({3, 4}; 204). The requisite {3, 4}-DGDD is constructed on Z204 with groups {i, 4 + i, . . . , 200 + i}, i = 0, 1, 2, 3 and holes {j, 17 + j, 34 + j, . . . , 187 + j}, j = 0, 1, . . . , 16. The design is obtained by developing the following base blocks +2 (mod 204). For r4 = 32:

{3, 124, 177}a {5, 190, 203}b {0, 39, 46}c {0, 177, 86}d {4, 2, 141, 187} {2, 65, 143, 76} {11, 90, 60, 193} {2, 201, 111, 100}

{0, 94, 181}a {8, 18, 175}b {11, 61, 2}c {3, 166, 4}d {11, 1, 94, 88} {7, 188, 61, 106} {7, 114, 48, 129} {8, 101, 174, 115}

{8, 30, 79}a {9, 62, 124}b {7, 113, 112}c {1, 95, 92}d {11, 80, 26, 149} {0, 186, 129, 191} {2, 203, 117, 156} {3, 21, 46, 180}

{2, 197, 35}a {3, 169, 132}b {9, 42, 188}c {5, 139, 66}d {2, 128, 171, 97} {8, 15, 73, 198} {1, 148, 174, 203} {11, 180, 37, 66}.

Each block of size 4 generates two parallel classes and the blocks of size 3 with the same superscript generate six parallel classes. For r4 = 36:

{4, 181, 51}a {3, 149, 158}b {2, 177, 135}c {8, 21, 47, 126} {7, 125, 166, 144} {3, 33, 144, 78} {9, 198, 131, 72} {7, 134, 77, 12}

{9, 19, 110}a {1, 130, 36}b {11, 178, 0}c {2, 151, 37, 16} {6, 31, 189, 8} {9, 172, 59, 142} {11, 137, 108, 10} {11, 33, 2, 48}.

{5, 120, 30}a {9, 107, 160}b {4, 138, 77}c {2, 200, 167, 1} {2, 113, 111, 56} {9, 152, 71, 170} {6, 13, 123, 44}

{8, 95, 154}a {6, 68, 199}b {1, 44, 139}c {3, 6, 16, 9} {7, 192, 142, 21} {11, 118, 76, 29} {11, 65, 114, 188}

Each block of size 4 generates two parallel classes and the blocks of size 3 with the same superscript generate six parallel classes. For r4 = 38:

{47, 61, 164}a {46, 96, 49}c {94, 149, 92}e {48, 59, 137, 78} {191, 29, 86, 192} {0, 1, 75, 114} {164, 103, 42, 81} {96, 10, 81, 143}.

{60, 171, 34}a {146, 101, 183}c {0, 109, 195}e {64, 46, 53, 23} {182, 144, 7, 5} {13, 102, 171, 196} {79, 158, 145, 200}

{1, 156, 51}b {9, 96, 175}d {103, 88, 121, 26} {191, 93, 68, 62} {49, 98, 103, 156} {0, 45, 66, 139} {20, 98, 161, 151}

{32, 22, 167}b {89, 80, 94}d {15, 116, 85, 42} {57, 31, 60, 38} {54, 11, 108, 17} {100, 123, 30, 65} {191, 178, 101, 20}

Each block of size 4 generates two parallel classes and the blocks of size 3 with the same superscript generate three parallel classes.  Lemma 3.11. All admissible URDs({3, 4}; 204) exist. Proof. For r4 ∈ {1, 3, 5, 7, 9}, see Theorem 1.2. For r4 ∈ {11, 13, 15, . . . , 47, 49, 53, 55}, the designs have been constructed in Lemmas 3.9 and 3.10. For r4 = 51, take a 4-RGDD of type 514 from Theorem 2.5 and fill in each group with an RPBD({3}; 51). For r4 ∈ {57, 59, 61, 63}, start with a {3, 4}-IUGDD of type (24, 0)7 (36, 36)1 with r4 = 56 and r30 = 6 from Lemma 3.8 and fill in the groups. For r4 ∈ {65, 67}, see Theorem 1.3.  Lemma 3.12. All admissible URDs({3, 4}; 228) exist.

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1077

Proof. We start with a 4-frame of type 35 from Theorem 2.1. Assign each point weight 15, and for each holey parallel class Pi replace the blocks in it with {3, 4}-URGDDs of type 154 with ni 4-pcs, ni ∈ {1, 3, . . . , 15} (see [24, Lemma 2.5]). This yields a 45−3ni

uniform {3, 4}-frame of type i=1 (45; 3 2 4ni )1 . Then adjoin three ideal points and fill in each group with a {3, 4}-URGDD of type 316 with ni 4-pcs (obtained from a URD({3, 4}, 48) by removing a 3-pc). Then we obtain a {3, 4}-URGDD of type 376 with r4 ∈ {5, 7, . . . , 75}, which can be regarded as a URD({3, 4}, 228) with the same r4 . For r4 ∈ {1, 3}, see Theorem 1.2. 

5

Lemma 3.13. There exists a URD({3, 4}; 264) with r4 = 79. Proof. We start with a {3, 4}-URGDD of type 2411 with r4 = 72 and fill in each group with a URD({3, 4}; 24) with r4 = 7 to obtain the desired design. The requisite URGDD is constructed on X = Z132 × {0, 1} with groups {{i, i + 11, i + 22, . . . , i + 121} × {0, 1}}, i = 0, 1, 2, . . . , 10. Develop the following base blocks (+1 (mod 132), −) and (−, +1 (mod 2)). Note that each of the blocks marked ∗ generates four 4-pcs, the blocks marked a generate forty-four 4-pcs and the blocks marked b generate twelve 3-pcs.

{(12, 0), (10, 1), (101, 1), (99, 1)}∗ {(7, 0), (110, 1), (37, 1), (100, 1)}∗ {(6, 0), (117, 0), (67, 0), (32, 1)}∗ {(8, 0), (3, 0), (81, 1), (126, 1)}∗ {(16, 0), (45, 0), (40, 1), (9, 0)}a {(106, 0), (42, 0), (55, 1), (103, 1)}a {(82, 0), (70, 1), (63, 0), (66, 0)}a {(13, 0), (61, 1), (21, 1), (93, 0)}a {(122, 0), (52, 0), (88, 1), (46, 0)}a {(28, 0), (45, 1), (27, 1)}b {(102, 0), (67, 0), (44, 0)}b

{(7, 0), (34, 1), (125, 0), (72, 1)}∗ {(7, 0), (65, 1), (82, 1), (88, 0)}∗ {(12, 0), (33, 1), (119, 0), (82, 1)}∗ {(28, 0), (4, 0), (24, 0), (36, 0)}a {(98, 0), (79, 1), (51, 0), (23, 0)}a {(73, 0), (58, 1), (74, 0), (127, 0)}a {(27, 0), (119, 1), (47, 1), (129, 0)}a {(33, 0), (131, 0), (64, 0), (113, 1)}a {(37, 0), (50, 0), (76, 0), (12, 1)}a {(14, 0), (131, 0), (41, 0)}b {(0, 0), (37, 0), (46, 0)}b . 

Lemma 3.14. There exists an IURD({3, 4}; 12n, 12) for 1. n ≥ 3, r3 = 6(n − 1), r4 = 0, r30 = 4 and r40 = 1; 2. n ≥ 4, r3 = 0, r4 = 4(n − 1), r30 = 4 and r40 = 1.

Proof. We start with a 3-RGDD of type 12n or a 4-RGDD of the same type from Theorem 2.5. Apply Construction 2.2 and fill in each of the groups except the last one with a URD({3, 4}; 12) to obtain the desired design.  Lemma 3.15. There exists an IURD({3, 4}; 60, 12) with 1. r3 = 24, r4 = 0, r30 = 4 and r40 = 1; 2. r3 = 18, r4 = 4, r30 = 4 and r40 = 1; 3. r3 = 6, r4 = 12, r30 = 4 and r40 = 1; 4. r3 = 0, r4 = 16, r30 = 4 and r40 = 1. Proof. For Cases 1 and 4, see Lemma 3.14. For Case 2, start with an RTD(4, 5) from Theorem 2.5 and redefine the groups to get a {4, 5}-URGDD of type 45 with r4 = 4 and r5 = 1. Assign each point with weight 3 and replace the blocks with {3, 4}-URGDDs of type 34 or 3-RGDDs of type 35 . Then we get a {3, 4}-URGDD of type 125 with r4 = 4. Fill in each of the groups except the last one with a URD({3, 4}; 12) to obtain the desired design. For Case 3, we first construct a resolvable 4-DGDD of type (12, 34 )5 on Z60 with groups {i, i + 5, . . . , 55 + i}, i = 0, 1, . . . , 4 and holes {j, j + 4, . . . , 56 + j}, j = 0, 1, 2, 3. The design is obtained by developing the following blocks +2 (mod 60). Note that each base block generates two parallel classes.

{48, 14, 17, 31}

{1, 12, 14, 23}

{31, 34, 52, 25}

{1, 19, 42, 48}

{0, 1, 38, 59}

{0, 7, 14, 41}.

Now, fill in the holes of this DGDD with 3-RGDDs of type 35 to obtain a {3, 4}-URGDD of type 125 with r4 = 12. Then fill in each of the groups except the last one with a URD({3, 4}; 12) to obtain the desired design.  Lemma 3.16. All admissible URDs({3, 4}; 348) and URD({3, 4}; 396) exist. Proof. For v = 348 and 1 ≤ r4 ≤ 113, we start from a TD(6, 7) and truncate two of its groups to obtain a TD(4, 7) with two resolutions P and Q, such that for each parallel class Pi ∈ P and each parallel class Qj ∈ Q, |Pi ∩ Qj | = 1. Pick a class Q and remove all the blocks in it to obtain a 4-MGDD of type 47 with each class Pi′ = Pi \Q forming a holey parallel class.

1078

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

Assign each point with weight 12, and for each Pi′ replace the blocks of it with {3, 4}-URGDDs of type 124 with ni 4-pcs, ni ∈ {0, 2, . . . , 12} (see [24, Lemma 2.3]). This yields a uniformly frame resolvable {3, 4}-DGDD of type

(48, 124 ; 412 )x (48, 124 ; 318 )6−x (48, 124 ; 3

36−3y 2

4y )1

with 0 ≤ x ≤ 6 and y ∈ {0, 2, 4, . . . , 12}. Now, we can fill in the holes of this DGDD with 3-frames of type 127 or 4-frames of the same type (see Theorem 2.1). 1. If we fill in the holes with 3-frames, we obtain a {3, 4}-frame of type

(48; 36 412 )x (48; 324 )6−x (48; 3

48−3y 2

4y )1 .

Adjoin 12 ideal points and fill in the last group with a URD({3, 4}; 60) with r4 = y + 1 (see Theorem 3.7) and fill in the other groups with IURD({3, 4}; 60, 12) with (r4 , r40 ) = {(0, 1), (12, 1)} (see Lemma 3.15). This yields a URD({3, 4}; 348) with 1 ≤ r4 = 12x + y + 1 ≤ 85. 2. If we fill in the holes with 4-frames, we obtain a {3, 4}-frame of type

(48; 416 )x (48; 318 44 )6−x (48; 3

36−3y 2

4y+4 )1 .

Adjoin 12 ideal points and fill in the last group with a URD({3, 4}; 60) with r4 = y + 5 (see Theorem 3.7) and fill in the other groups with IURD({3, 4}; 60, 12) with (r4 , r40 ) = {(4, 1), (16, 1)} (see Lemma 3.15). This yields a URD({3, 4}; 348) with 29 ≤ r4 = 12x + y + 29 ≤ 113. For v = 396 and 1 ≤ r4 ≤ 129, we proceed similarly, starting instead with a TD(6, 8). Finally, for v = 348 and r4 = 115, or v = 396 and r4 = 131, see Theorem 1.3.  4. Admissible URD({3, 4}; v) for v ≡ 0 (mod 24) Lemma 4.1. There exists a uniform {3, 4}-frame of type (6; 42 )u (6; 33 )5−u for each u ∈ {0, 1, 2, 3, 4, 5}. Proof. For u = 0 or 5, see Theorem 2.1. For u = 4, the required frame is constructed on X = Z3 × {0, 1, 2, . . . , 9} with groups Z3 × {i, i + 5} for i = 0, 1, 2, 3, 4. Develop the following base blocks (+1 (mod 3), −). Note that all the eight blocks of size 3 form a holey parallel class associated with the group Z3 × {0, 5} and we can get three holey 3-pcs when they are developed. Note also that the two blocks of size four in each row generate a holey parallel class.

{01 , 02 , 03 } {06 , 04 , 12 } {03 , 05 , 14 , 17 } {00 , 07 , 14 , 13 } {06 , 19 , 15 , 03 } {24 , 21 , 18 , 25 } {04 , 22 , 26 , 00 } {24 , 05 , 22 , 06 } {11 , 25 , 03 , 22 } {00 , 03 , 12 , 21 }

{13 , 26 , 19 } {07 , 09 , 21 } {20 , 18 , 19 , 22 } {19 , 25 , 08 , 02 } {01 , 24 , 00 , 08 } {10 , 03 , 26 , 29 } {15 , 21 , 17 , 29 } {00 , 17 , 11 , 09 } {26 , 17 , 20 , 08 } {05 , 08 , 16 , 27 }

{08 , 22 , 29 } {11 , 24 , 28 }

{18 , 17 , 16 } {27 , 14 , 23 }

For u = {1, 2, 3}, the designs are constructed on Z6 × {0, 1, 2, 3, 4} with groups Z6 × {i}, i = 0, 1, 2, 3, 4. Develop the following base blocks (+1 (mod 6), −). Note that the blocks in each row generate all the holey parallel classes associated with the same group. u=3:

{01 , 02 , 03 } {23 , 54 , 20 } {01 , 44 , 53 , 30 } {20 , 31 , 04 , 42 } {01 , 23 , 52 , 10 }

{21 , 42 , 14 } {02 , 13 , 14 } {13 , 34 , 30 , 31 } {24 , 30 , 42 , 11 } {00 , 02 , 21 , 53 }

{04 , 41 , 53 } {00 , 33 , 52 }

{13 , 52 , 54 } {04 , 12 , 40 }

u=2:

{13 , 41 , 14 } {20 , 53 , 02 } {04 , 51 , 13 } {00 , 52 , 24 , 31 } {00 , 22 , 03 , 11 }

{22 , 53 , 04 } {00 , 12 , 34 } {11 , 30 , 23 } {00 , 02 , 04 , 21 } {23 , 12 , 41 , 40 }

{01 , 02 , 03 } {03 , 24 , 40 } {03 , 50 , 44 }

{24 , 21 , 12 } {13 , 44 , 52 } {24 , 01 , 10 }

u=1:

{01 , 02 , 03 } {02 , 54 , 43 } {21 , 24 , 43 } {12 , 01 , 14 } {10 , 22 , 01 , 53 }

{22 , 44 , 51 } {12 , 30 , 33 } {11 , 50 , 23 } {11 , 52 , 30 } {00 , 02 , 11 , 53 }. 

{11 , 34 , 43 } {04 , 22 , 50 } {10 , 33 , 04 } {24 , 51 , 20 }

{12 , 23 , 24 } {23 , 44 , 10 } {00 , 01 , 44 } {10 , 34 , 02 }

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1079

Theorem 4.2. There exist all admissible URDs({3, 4}; v) for v ≡ 0 (mod 24). Proof. Combining Theorem 1.5, Lemmas 3.3 and 3.13, there only remain the cases where v ∈ {408, 456, 552, 984, 1128, 3288} and v3 − 15 ≤ r4 ≤ v3 − 9 to be considered.

For v = 408 and 121 ≤ r4 ≤ 127, write r4 = 121 + y with y ∈ {0, 2, 4, 6}. We start with a 5-GDD of type 611 from Theorem 2.6 and apply Construction 2.4, assigning each point with weights (6; 30 42 ) or (6; 33 40 ). Note that there exist uniform {3, 4}-frames of type (6; 42 )u (6; 33 )5−u with u ∈ {0, 1, 2, 3, 4, 5} by Lemma 4.1. We can obtain a uniform {3, 4}36−3y

frame of type (36; 412 )10 (36; 3 2 4y )1 . Then adjoin 12 ideal points and apply Construction 2.3; fill in the last group with a URD({3, 4}; 48) with r4 = y + 1 (see Theorem 3.7) and fill in the other groups with IURD({3, 4}; 48, 12) (see Lemma 3.14). This yields a URD({3, 4}; 408) with r4 = 121 + y, as desired. For v ∈ {552, 3288}, the construction is exactly the same as that for v = 408. We start with 5-GDDs of types 185 or 21 26 respectively (see Theorem 2.6). Assign each point with weights (6; 30 42 ) or (6; 33 40 ) to get a uniform {3, 4}-frame. Then adjoin 12 ideal points and fill in the groups. The input URD({3, 4}; 120), URD({3, 4}; 168), IURD({3, 4}; 120, 12) and IURD({3, 4}; 168, 12) come from Theorem 3.7 and Lemma 3.14. For v = 456 and 137 ≤ r4 ≤ 143, write r4 = 136 + z with z ∈ {1, 3, 5, 7}. We start with a 5-GDD of type 126 from Theorem 2.6 and apply Construction 2.4, assigning each point with weights (6; 30 42 ) or (6; 33 40 ). That gives us a uniform {3, 4}-frame of type (72; 424 )5 (72; 312 416 )1 . Then adjoin 24 ideal points and fill in the first five groups with 4-RGDDs of type 244 (see Theorem 2.5). Then we get a {3, 4}-IUGDD of type (24, 0)15 (96, 96)1 with r4 = 120 and r40 = 16. Fill in each group of size 24 with a URD({3, 4}; 24) with z 4-pcs and fill in the group of size 96 with a URD({3, 4}; 96) with 16 + z 4-pcs (Theorem 3.7). Thus we get a URD({3, 4}; 456) with r4 = 136 + z, as desired. For v = 984 or 1128, we proceed similarly, starting instead with 5-GDDs of types 325 (Theorem 2.6) or 325 241 (Theorem 2.7) to obtain the desired designs. The input 4-RGDD of type 249 comes from Theorem 2.5, and the input URD({3, 4}; 216) and URD({3, 4}; 168) come from Theorem 1.5.  5. Admissible URD({3, 4}; v) for v ≡ 12 (mod 24) Lemma 5.1. Let n ≥ 1 be a positive integer. Then all admissible URDs({3, 4}; 120n + 12) exist. Proof. For n = 1, see Theorem 3.7. For r4 = 40n + 3, see Theorem 1.3. For n ≥ 5 and 1 ≤ r4 ≤ 40n + 1, write r4 = 40x + y + 1 with 0 ≤ x ≤ n − 1 and 0 ≤ y ≤ 40. There exists a 5-GDD of type 20n by Theorem 2.6. Assign weights (6; 30 42 ) or (6; 33 40 ) to each point of this GDD and apply Construction 2.4. Since there exist uniform {3, 4}-frames of type (6; 42 )u (6; 33 )5−u with u ∈ {0, 1, 2, 3, 4, 5} by Lemma 4.1, 120−3y

we can obtain a uniform {3, 4}-frame of type (120; 440 )x (120; 360 )n−1−x (120; 3 2 4y )1 . Then adjoin 12 ideal points and apply Construction 2.3; fill in the last group with a URD({3, 4}; 132) with r4 = y + 1 (see Theorem 3.7) and fill in the other groups with IURD({3, 4}; 132, 12) with r4 = 0 or 40 and r40 = 1 (see Lemma 3.14). This yields a URD({3, 4}; 120n + 12) with r4 = 40x + y + 1, as desired. For n ∈ {2, 3, 4} and 1 ≤ r4 ≤ 40n + 1, write r4 = 8nx + y + 1 with 0 ≤ x ≤ 4 and 0 ≤ y ≤ 8n. Start with a TD(5, 4n) from Theorem 2.8 and assign each point with weights (6; 30 42 ) or (6; 33 40 ) to obtain a uniform {3, 4}24n−3y

frame of type (24n; 48n )x (24n; 312n )4−x (24n; 3 2 4y )1 . Then adjoin 12 ideal points and fill in the groups to obtain a URD({3, 4}; 120n + 12) with r4 = 8nx + y + 1, as desired; the input URD({3, 4}; 24n + 12) and IURD({3, 4}; 24n + 12, 12) come from Theorem 3.7 and Lemma 3.14.  Lemma 5.2. There exists an IURD({3, 4}; 156, 36) with r3 = 60, r4 = 0, r30 = 16 and r40 = 1, and an IURD({3, 4}; 156, 36) with r3 = 0, r4 = 40, r30 = 16 and r40 = 1. Proof. For the case of r3 = 60 and r4 = 0, start with a 3-frame of type 246 from Theorem 2.5, adjoin 12 ideal points and fill in all the groups except the last one with 3-RGDDs of type 123 . This yields a 3-IUGDD of type (12, 0)10 (36, 36)1 with r3 = 60 and r30 = 12. Then fill in each group of size 12 with a URD({3, 4}; 12) to obtain an IURD({3, 4}; 156, 36) with r3 = 60, r4 = 0, r30 = 16 and r40 = 1. For the case of r3 = 0 and r4 = 40, start with a {3, 4}-IUGDD of type (24, 0)5 (36, 36)1 with r4 = 40 and r30 = 6 from Lemma 3.5 and fill in each group of size 24 with a URD({3, 4}; 24) with r3 = 10 and r4 = 1 to obtain the desired design.  Lemma 5.3. Let n be a nonnegative integer. Then all admissible URDs({3, 4}; 120n + 36) exist. Proof. For n ∈ {0, 1, 3}, see Theorem 3.7 and Lemma 3.16. For r4 = 40n + 11, see Theorem 1.3. For n ≥ 2, n ̸= 3 and 40n + 1 ≤ r4 ≤ 40n + 9, write r4 = 40n + y + 1 with y ∈ {0, 2, 4, 6, 8}. Start with a 5-GDD of type (4n)5 41 from Theorem 2.7 and assign each point with weights (6; 30 42 ) or (6; 33 40 ) to obtain a uniform {3, 4}-frame 24−3y

of type (24n; 48n )5 (24; 3 2 4y )1 . Then adjoin 12 ideal points, fill in the last group with a URD({3, 4}; 36) with r4 = y + 1 and fill in all the other groups with IURD({3, 4}; 24n + 12, 12) with r4 = 8n and r40 = 1. That gives a URD({3, 4}; 120n + 36) with r4 = 40n + y + 1, as desired.

1080

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

For n ≥ 5 and 1 ≤ r4 < 40n + 1, write r4 = 40x + y + 1 with 0 ≤ x ≤ n − 1 and 0 ≤ y < 40. Start with a 5-GDD of type 20n and assign each point with weights (6; 30 42 ) or (6; 33 40 ) to obtain a uniform {3, 4}-frame of type 120−3y

(120; 440 )x (120; 360 )n−1−x (120; 3 2 4y )1 . Then adjoin 36 ideal points and apply Construction 2.3; fill in the last group with a URD({3, 4}; 156) with r4 = y + 1 (see Theorem 3.7) and fill in the other groups with IURD({3, 4}; 156, 36) with r4 = 0 or 40 and r40 = 1 (see Lemma 5.2). This yields a URD({3, 4}; 120n + 36) with r4 = 40x + y + 1, as desired. For n ∈ {2, 4} and 1 ≤ r4 < 40n + 1, write r4 = 8nx + y + 1 with 0 ≤ x ≤ 4 and 0 ≤ y ≤ 8n. Start from a 5-GDD of type (4n)5 41 and assign each point with weights (6; 30 42 ) or (6; 33 40 ) to obtain a uniform {3, 4}-frame of type (24n; 48n )x (24n; 312n )4−x (24n; 3

24n−3y 2

4y )1 (24; 312 )1 .

Then adjoin 12 ideal points and fill in the groups to obtain a URD({3, 4}; 120n+36) with r4 = 8nx+y+1, as desired; the input URD({3, 4}; 24n + 12), IURD({3, 4}; 36, 12) and IURD({3, 4}; 24n + 12, 12) come from Theorem 3.7 and Lemma 3.14.  Lemma 5.4. All admissible URDs({3, 4}; v) exist for v = 120n + 84 or 120n + 108 with n ≡ 0, 1, 2 (mod 5). Proof. For v ∈ {84, 108}, see Theorem 3.7. For v ∈ {204, 228, 348}, see Lemmas 3.11, 3.12 and 3.16. For v = 324, see [24, Theorem 5.1]. For v = 120n + 84 with 1 ≤ r4 ≤ 40n + 25 and n ≥ 5, write r4 = 8nx + y + z + 1 with 0 ≤ x ≤ 4, 0 ≤ y ≤ 8n and z ∈ {0, 24}. Start from a 5-GDD of type (4n)5 121 and assign each point with weights (6; 30 42 ) or (6; 33 40 ) to obtain a uniform {3, 4}-frame of type

(24n; 48n )x (24n; 312n )4−x (24n; 3

24n−3y 2

4y )1 (72; 3

72−3z 2

4z )1 .

Then adjoin 12 ideal points and fill in the groups to obtain a URD({3, 4}; 120n + 84) with r4 = 8nx + y + z + 1, as desired; the input URD({3, 4}; 24n + 12) for n ≡ 0, 1, 2 (mod 5) comes from Lemmas 5.1 and 5.3 and Theorem 1.6, respectively, and the IURD({3, 4}; 84, 12) and IURD({3, 4}; 24n + 12, 12) come from Lemma 3.14. For v = 120n + 108 with 1 ≤ r4 ≤ 40n + 33 and n ≥ 5, we proceed similarly. Write r4 = 8nx + y + z + 1 with 0 ≤ x ≤ 4, 0 ≤ y ≤ 8n and z ∈ {0, 32}. Start from a 5-GDD of type (4n)5 161 and assign each point with weights (6; 30 42 ) or (6; 33 40 ). Then adjoin 12 ideal points and fill in the groups to obtain the desired designs. Finally, for v = 120n + 84 and r4 = 40n + 27, or v = 120n + 108 and r4 = 40n + 35, see Theorem 1.3.  Lemma 5.5. All admissible URDs({3, 4}; v) exist for v = 120n + 84 or 120n + 108 with n ≡ 3 (mod 5). Proof. For v = 120n + 84 and r4 = 40n + 27, or v = 120n + 108 and r4 = 40n + 35, see Theorem 1.3. For v = 120n + 84 with n ≥ 8 and 1 ≤ r4 ≤ 40n + 25, write r4 = (8n − 8)x + y + z + 1 with 0 ≤ x ≤ 4, 0 ≤ y ≤ 8n − 8 and z ∈ {0, 64}. Start with a 5-GDD of type (4n − 4)5 321 from Theorem 2.7 and assign each point with weights (6; 30 42 ) or (6; 33 40 ) to obtain a uniform {3, 4}-frame of type

(24n − 24; 48n−8 )x (24n − 24; 312n−12 )4−x (24n − 24; 3

24n−3y−24 2

4y )1 (192; 3

192−3z 2

4z )1 .

Then adjoin 12 ideal points and fill in the groups to obtain a URD({3, 4}; 120n + 84) with r4 = (8n − 8)x + y + z + 1, as desired; the input URD({3, 4}; 24n − 12), IURD({3, 4}; 204, 12) and IURD({3, 4}; 24n − 12, 12) come from Theorem 1.6 and Lemma 3.14. For v = 120n + 108 with n ≥ 8 and 1 ≤ r4 ≤ 40n + 33, we proceed similarly. Write r4 = (8n − 8)x + y + z + 1 with 0 ≤ x ≤ 4, 0 ≤ y ≤ 8n − 8 and z ∈ {0, 72}. Start from a 5-GDD of type (4n − 4)5 361 and assign each point with weights (6; 30 42 ) or (6; 33 40 ). Then adjoin 12 ideal points and fill in the groups to obtain the desired designs. For v = 444 or 468, we start with a 5-GDD of type 126 or type 125 161 (Theorem 2.7) and assign each point weights (6; 30 42 ) or (6; 33 40 ). Then adjoin 12 ideal points and fill in the groups to obtain the desired designs.  Lemma 5.6. All admissible URDs({3, 4}; v) exist for v = 120n + 84 or 120n + 108 with n ≡ 4 (mod 5). Proof. For v = 120n + 84 and r4 = 40n + 27, or v = 120n + 108 and r4 = 40n + 35, see Theorem 1.3. For v = 120n + 84 with n ≥ 9 and 1 ≤ r4 ≤ 40n + 25, write r4 = 40x + y + z + 1 with 0 ≤ x ≤ n − 1, 0 ≤ y ≤ 40 and z ∈ {0, 24}. Start with a TD(6, n) from Theorem 2.8, adjoin an ideal point and remove an original point to redefine the groups to obtain a {6, n + 1}-GDD of type 5n n1 . Assign weight 4 to the ideal point and the points in the first n groups, and assign weights 0 or 4 to the remaining points. Note that there exist 5-GDDs of types 45 , 46 and 4n+1 by Theorem 2.6. We obtain a 5-GDD of type 20n 121 . Then assign each point with weights (6; 30 42 ) or (6; 33 40 ) to obtain a uniform {3, 4}-frame of type

(120; 440 )x (120; 360 )n−1−x (120; 3

120−3y 2

4y )1 (72; 3

72−3z 2

4z )1 .

Adjoin 12 ideal points and fill in the groups to obtain a URD({3, 4}; 120n + 84) with r4 = 40x + y + z + 1, as desired; the input URD({3, 4}; 132), IURD({3, 4}; 84, 12) and IURD({3, 4}; 132, 12) come from Theorem 3.7 and Lemma 3.14. For v = 120n + 108 with n ≥ 9 and 1 ≤ r4 ≤ 40n + 33, we proceed similarly. Write r4 = 40x + y + z + 1 with 0 ≤ x ≤ n − 1, 0 ≤ y ≤ 40 and z ∈ {0, 32}. Start with the {6, n + 1}-GDD of type 5n n1 constructed above and assign weights

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1081

0 or 4 to obtain a 5-GDD of type 20n 161 . Then assign each point weights (6; 30 42 ) or (6; 33 40 ) to get a uniform {3, 4}-frame of type

(120; 440 )x (120; 360 )n−1−x (120; 3

120−3y 2

4y )1 (96; 3

96−3z 2

4z )1 .

Then adjoin 12 ideal points and fill in the groups to obtain a URD({3, 4}; 120n + 108) with r4 = 40x + y + z + 1, as desired. For v = 564 or 588, we start with a 5-GDD of type 165 121 or type 166 (Theorem 2.7) and assign each point weights (6; 30 42 ) or (6; 33 40 ). Then adjoin 12 ideal points and fill in the groups to obtain the desired designs.  Combining the results in this section together with Theorem 1.6, we obtain the following results. Theorem 5.7. All admissible URDs({3, 4}; v) exist for v = 12 (mod 24), except for v = 12 and r4 = 3. Acknowledgments First author’s research was supported by China Postdoctoral Science Foundation under Grant No. 2015M571067 and Beijing Postdoctoral Research Foundation. Second author’s research was supported by the National Natural Science Foundation of China under Grant Nos. 61171198, 11431003 and 61571310, and the Importation and Development of HighCaliber Talents Project of Beijing Municipal Institutions. Appendix Each of the following {3, 4}-GDDs of type 514 is constructed on Z204 with groups {i, 4 + i, . . . , 200 + i}, i = 0, 1, 2, 3. The design is obtained by developing the elements of Z204 in the given base blocks +2 (mod 204), together with the blocks {i, 51 + i, 102 + i, 153 + i}, i = 0, 1, . . . , 50. r4 = 11:

{4, 73, 134} {1, 163, 22} {3, 42, 157} {3, 96, 141} {2, 61, 99} {3, 97, 172} {5, 52, 183} {0, 47, 10, 25}

{4, 86, 195} {4, 110, 87} {3, 62, 188} {2, 97, 164} {2, 191, 93} {3, 152, 49} {3, 80, 130} {0, 201, 11, 198}

{3, 94, 164} {4, 153, 83} {2, 199, 125} {4, 163, 197} {4, 129, 170} {4, 70, 155} {2, 144, 73} {0, 199, 13, 190}.

{39, 200, 86} {37, 76, 67} {41, 122, 167} {31, 48, 174} {32, 78, 137} {32, 107, 165} {0, 137, 171} {10, 57, 35, 20}.

{37, 50, 99} {33, 139, 52} {38, 169, 123} {42, 184, 115} {36, 179, 113} {40, 179, 25} {11, 14, 25, 8}

{2, 109, 28} {4, 143, 162} {1, 151, 84} {2, 60, 121} {1, 143, 38} {3, 34, 120} {0, 73, 195}

{2, 31, 157} {4, 154, 53} {3, 89, 132} {3, 20, 114} {2, 55, 24} {3, 108, 197} {0, 7, 2, 1}

{1, 88, 59} {3, 93, 36} {4, 139, 174} {1, 58, 31} {5, 30, 144} {1, 142, 172} {0, 23, 21, 186}

r4 = 13:

{41, 83, 146} {34, 97, 87} {35, 62, 157} {35, 58, 168} {43, 118, 192} {34, 179, 69} {37, 138, 171} {10, 9, 8, 15}

{35, 125, 6} {42, 80, 159} {36, 106, 197} {41, 90, 144} {30, 52, 145} {35, 61, 92} {10, 3, 41, 188}

{40, 107, 162} {40, 161, 186} {33, 183, 86} {37, 128, 78} {36, 161, 2} {36, 147, 66} {10, 19, 37, 24}

{39, 154, 68} {39, 50, 116} {37, 167, 58} {33, 130, 172} {45, 180, 82} {40, 123, 5} {10, 28, 51, 49}

r4 = 15:

{5, 64, 167} {5, 164, 99} {4, 58, 173} {5, 139, 22} {7, 200, 122} {5, 163, 62} {1, 7, 2, 0} {1, 140, 162, 195}.

{4, 77, 167} {6, 140, 87} {5, 110, 156} {6, 179, 120} {5, 98, 19} {8, 95, 149} {1, 168, 183, 158}

{6, 148, 73} {7, 56, 161} {4, 165, 90} {4, 134, 87} {5, 132, 38} {6, 77, 143} {1, 167, 174, 148}

{4, 150, 127} {5, 131, 192} {2, 167, 36} {7, 150, 185} {6, 72, 191} {6, 113, 199} {1, 4, 131, 202}

{6, 35, 44} {8, 114, 101} {4, 113, 83} {4, 158, 89} {6, 189, 155} {7, 90, 153} {1, 184, 166, 3}

{4, 178, 163} {5, 67, 96} {4, 73, 171} {5, 127, 32} {5, 160, 118} {0, 82, 179} {1, 178, 187, 192}

1082

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

r4 = 17:

{0, 189, 123} {0, 173, 86} {1, 71, 90} {0, 157, 107} {0, 109, 171} {1, 27, 112} {0, 18, 41, 39}

{0, 169, 30} {2, 148, 107} {0, 133, 62} {0, 69, 34} {0, 193, 66} {0, 49, 175} {0, 43, 182, 33}

{1, 43, 92} {1, 95, 138} {2, 123, 181} {0, 117, 54} {0, 183, 9} {0, 95, 162} {0, 26, 57, 19}

{1, 102, 56} {0, 181, 90} {0, 70, 143} {1, 99, 126} {0, 110, 97} {0, 53, 99} {0, 37, 15, 194}

{1, 64, 123} {0, 167, 82} {1, 6, 131} {0, 81, 126} {1, 35, 104} {0, 3, 6, 17} {0, 83, 38, 29}

{0, 147, 129} {1, 40, 115} {1, 18, 116} {1, 128, 54} {1, 144, 94} {0, 2, 1, 7} {0, 14, 27, 145}.

r4 = 19:

{3, 61, 86} {3, 45, 104} {2, 97, 164} {3, 102, 56} {4, 134, 85} {1, 27, 120} {1, 15, 202, 4}

{2, 125, 64} {4, 121, 167} {4, 74, 151} {2, 80, 133} {5, 174, 111} {0, 93, 106} {1, 164, 203, 182}

{2, 172, 127} {3, 38, 125} {1, 175, 48} {3, 42, 157} {3, 88, 69} {1, 180, 23, 190} {1, 19, 6, 196}

{2, 135, 73} {2, 177, 88} {4, 126, 71} {2, 156, 61} {1, 131, 34} {1, 95, 156, 126} {0, 203, 5, 202}

{3, 138, 117} {3, 137, 30} {3, 134, 68} {2, 116, 93} {3, 46, 121} {1, 140, 162, 195} {1, 151, 122, 160}.

{4, 114, 103} {3, 18, 81} {4, 154, 117} {3, 106, 37} {5, 80, 22} {0, 197, 31, 178}

r4 = 21:

{6, 119, 97} {8, 90, 179} {5, 90, 67} {4, 50, 159} {8, 195, 70} {1, 203, 182, 164} {1, 198, 200, 199}

{6, 77, 195} {4, 167, 114} {5, 108, 18} {7, 38, 153} {2, 159, 56} {2, 31, 40, 85} {0, 105, 30, 79}

{6, 135, 72} {7, 152, 94} {4, 77, 143} {6, 129, 199} {6, 132, 21} {1, 94, 84, 131} {1, 167, 174, 148}

{7, 124, 85} {7, 53, 78} {4, 99, 158} {5, 72, 179} {5, 175, 26} {1, 196, 6, 19} {2, 45, 184, 35}.

{3, 93, 130} {6, 167, 73} {6, 175, 133} {7, 64, 161} {0, 61, 143} {2, 71, 36, 177}

{3, 76, 182} {6, 191, 92} {6, 136, 91} {5, 128, 58} {6, 141, 168} {2, 13, 200, 203}

{106, 173, 27} {96, 6, 69} {111, 136, 26} {93, 7, 150} {106, 33, 63} {26, 0, 57, 127} {28, 61, 71, 6}

{106, 61, 152} {93, 106, 176} {105, 144, 31} {104, 170, 85} {111, 192, 85} {26, 31, 24, 25} {25, 124, 91, 154}.

{100, 77, 178} {100, 129, 91} {87, 128, 33} {88, 41, 167} {0, 38, 149} {23, 154, 92, 137}

{116, 135, 202} {102, 201, 151} {101, 190, 132} {104, 93, 51} {27, 2, 49, 12} {28, 123, 157, 70}

r4 = 23:

{109, 38, 155} {103, 168, 14} {101, 18, 163} {106, 187, 32} {93, 190, 175} {27, 196, 162, 125} {25, 27, 4, 190}

{105, 134, 12} {97, 38, 3} {101, 24, 122} {83, 144, 90} {25, 14, 11, 8} {27, 32, 45, 18}

r4 = 25:

{91, 140, 1} {89, 190, 184} {95, 154, 121} {73, 191, 110} {93, 172, 143} {24, 69, 190, 15} {23, 65, 158, 192}

{95, 13, 98} {95, 28, 114} {88, 30, 15} {97, 110, 200} {0, 75, 98} {24, 19, 10, 37} {21, 4, 126, 7}.

{91, 138, 188} {81, 102, 8} {88, 22, 185} {83, 13, 24} {26, 0, 19, 57} {25, 192, 182, 3}

{84, 189, 38} {91, 166, 33} {94, 27, 105} {91, 162, 45} {23, 186, 0, 21} {26, 56, 105, 199}

{95, 161, 178} {93, 155, 182} {81, 162, 47} {97, 122, 184} {24, 186, 151, 77} {25, 90, 123, 68}

{87, 144, 198} {88, 151, 10} {96, 26, 139} {94, 39, 49} {25, 55, 168, 94} {25, 24, 26, 31}

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1083

r4 = 27:

{37, 175, 100} {37, 99, 140} {38, 148, 123} {39, 138, 196} {9, 199, 196, 202} {10, 93, 48, 39} {9, 192, 174, 11}.

{40, 9, 83} {35, 64, 118} {36, 134, 19} {38, 187, 137} {9, 98, 191, 176} {9, 186, 200, 195}

{36, 122, 195} {35, 116, 106} {37, 11, 144} {31, 44, 110} {11, 121, 182, 160} {9, 110, 144, 51}

{33, 110, 75} {46, 203, 104} {50, 185, 144} {33, 163, 202} {61, 68, 42, 155} {63, 190, 169, 156}

{151, 0, 166} {39, 150, 84} {45, 74, 163} {51, 78, 21} {61, 74, 175, 180} {63, 94, 184, 9}

{33, 54, 167} {38, 112, 27} {35, 161, 84} {34, 116, 69} {9, 126, 84, 179} {9, 156, 182, 175}

{36, 98, 21} {38, 185, 103} {35, 121, 54} {37, 47, 80} {9, 154, 104, 67} {10, 59, 184, 149}

{40, 17, 175} {37, 135, 110} {40, 13, 154} {0, 70, 151} {11, 12, 10, 17} {9, 122, 183, 76}

{57, 59, 190} {42, 116, 173} {17, 79, 168} {63, 4, 121, 158} {61, 56, 55, 54} {61, 127, 198, 144}

{47, 70, 148} {199, 44, 130} {56, 181, 118} {62, 40, 105, 95} {59, 48, 42, 45} {63, 29, 96, 138}.

r4 = 29:

{31, 154, 185} {24, 203, 165} {54, 188, 177} {55, 110, 96} {62, 101, 80, 19} {64, 171, 110, 45}

{41, 148, 175} {54, 199, 176} {38, 129, 83} {0, 9, 195} {65, 43, 18, 28} {68, 147, 173, 98}

r4 = 31:

{48, 111, 10} {50, 193, 87} {51, 142, 121} {43, 72, 150} {12, 130, 95, 41} {13, 51, 20, 198}

{52, 141, 194} {51, 129, 6} {51, 184, 145} {0, 14, 161} {13, 144, 59, 86} {12, 55, 194, 45}

{47, 62, 165} {45, 64, 199} {49, 160, 62} {12, 91, 117, 42} {11, 106, 56, 69} {13, 182, 55, 148}

{45, 23, 78} {52, 133, 11} {46, 37, 116} {14, 151, 60, 121} {13, 16, 10, 27} {14, 41, 23, 88}

{54, 43, 120} {43, 48, 133} {45, 190, 76} {15, 14, 16, 21} {12, 54, 141, 107} {14, 35, 200, 37}.

{43, 185, 68} {50, 149, 172} {44, 138, 185} {13, 192, 143, 202} {12, 79, 145, 162}

r4 = 33:

{20, 119, 106} {14, 79, 1} {3, 52, 66} {57, 66, 28, 111} {55, 138, 192, 121} {54, 73, 111, 80}

{6, 185, 88} {14, 140, 95} {186, 72, 165} {57, 27, 170, 124} {53, 59, 52, 54} {57, 4, 143, 166}

{15, 149, 48} {8, 197, 74} {8, 177, 47} {56, 74, 97, 99} {54, 89, 20, 131} {54, 84, 133, 159}

{9, 192, 98} {26, 173, 183} {10, 25, 140} {55, 44, 38, 41} {55, 37, 98, 28} {55, 110, 165, 132}.

{139, 21, 152} {146, 85, 120} {134, 89, 4} {37, 162, 63, 192} {38, 11, 96, 57} {35, 38, 32, 49}

{133, 176, 38} {135, 24, 9} {0, 155, 157} {40, 73, 18, 83} {37, 42, 28, 55} {36, 29, 130, 67}.

{146, 113, 20} {136, 83, 9} {136, 175, 30} {37, 188, 102, 71} {38, 88, 1, 147}

{157, 191, 28} {18, 183, 121} {19, 136, 38} {54, 64, 101, 199} {53, 7, 80, 138}

{33, 123, 38} {145, 123, 176} {0, 71, 193} {53, 82, 144, 3} {59, 0, 154, 117}

r4 = 35:

{142, 163, 28} {136, 33, 95} {137, 162, 55} {37, 60, 142, 131} {40, 75, 189, 6} {36, 183, 77, 54}

{128, 86, 5} {131, 89, 0} {36, 74, 65, 119} {38, 85, 63, 48} {36, 41, 35, 34}

{135, 8, 65} {130, 111, 200} {37, 104, 7, 150} {36, 7, 161, 98} {40, 161, 23, 94}

1084

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

r4 = 37:

{9, 126, 91} {0, 85, 114} {42, 161, 52} {41, 19, 94, 8} {38, 31, 160, 177} {37, 78, 167, 124}

{83, 109, 128} {28, 65, 30} {89, 47, 192} {32, 98, 169, 19} {29, 194, 11, 100} {30, 31, 136, 37}.

{67, 72, 10} {151, 184, 73} {34, 61, 196, 175} {33, 94, 80, 175} {34, 87, 84, 57}

{75, 44, 73} {128, 195, 125} {40, 22, 87, 101} {36, 6, 51, 5} {39, 98, 49, 64}

{3, 140, 41} {41, 108, 51} {5, 98, 75, 88} {13, 176, 155, 110} {11, 41, 126, 80}

{8, 55, 142} {1, 156, 30} {16, 195, 69, 102} {3, 14, 40, 173} {10, 65, 192, 119}

{13, 107, 88} {24, 163, 81} {17, 59, 136, 198} {4, 73, 174, 7} {13, 16, 74, 31}

{103, 20, 94} {18, 73, 139} {28, 47, 141, 178} {35, 198, 189, 176} {34, 157, 8, 195}

{54, 151, 60} {81, 120, 158} {34, 11, 181, 104} {35, 108, 133, 30} {32, 167, 81, 178}

r4 = 39:

{10, 179, 140} {23, 69, 128} {8, 69, 194, 71} {9, 50, 64, 59} {9, 16, 107, 126} {16, 27, 10, 41}.

{6, 65, 39} {18, 48, 91} {5, 4, 11, 6} {14, 49, 139, 64} {9, 184, 142, 95}

{17, 26, 124} {0, 13, 122} {10, 37, 183, 100} {12, 113, 162, 135} {16, 37, 54, 167}

r4 = 41:

{133, 194, 123} {128, 70, 169} {2, 143, 12, 173} {192, 143, 13, 14} {198, 183, 100, 105}

{167, 42, 20} {136, 133, 159} {0, 103, 21, 110} {1, 171, 82, 196} {3, 41, 144, 2}

{185, 51, 170} {116, 130, 133} {2, 97, 83, 52} {6, 75, 189, 40} {0, 158, 191, 185}

{141, 150, 203} {0, 6, 193} {6, 35, 88, 145} {202, 35, 17, 76} {0, 169, 42, 119}

{143, 77, 172} {196, 169, 191, 126} {3, 40, 42, 89} {199, 140, 93, 86} {203, 190, 57, 124}

{154, 124, 37} {4, 177, 15, 122} {6, 165, 44, 111} {2, 77, 188, 75} {0, 135, 181, 74}.

r4 = 43:

{180, 175, 57} {0, 5, 70} {78, 52, 75, 173} {82, 157, 91, 24} {91, 140, 9, 158}

{178, 12, 13} {184, 130, 175} {44, 162, 21, 131} {90, 27, 69, 96} {54, 145, 52, 71}

{165, 172, 51} {86, 200, 53, 39} {75, 45, 14, 60} {59, 132, 25, 70} {73, 79, 158, 52}

{158, 161, 99} {68, 46, 169, 11} {22, 33, 51, 152} {82, 187, 0, 53} {73, 178, 15, 148}

{148, 3, 158} {72, 97, 138, 119} {85, 122, 200, 31} {46, 80, 3, 157} {74, 61, 24, 139}.

{82, 151, 161} {81, 79, 134, 176} {65, 58, 152, 91} {70, 31, 69, 56}

r4 = 45:

{16, 74, 145} {192, 78, 175, 109} {191, 74, 17, 24} {194, 148, 25, 171} {9, 95, 148, 18}

{0, 13, 138} {31, 60, 46, 105} {14, 12, 21, 15} {18, 28, 133, 111} {172, 146, 75, 5}

{58, 4, 53} {194, 5, 60, 15} {184, 58, 159, 9} {29, 200, 91, 102} {8, 81, 14, 127}

{65, 4, 23} {13, 78, 16, 119} {2, 111, 1, 20} {109, 168, 190, 75} {18, 23, 5, 140}.

{133, 55, 8} {5, 3, 36, 66} {10, 180, 145, 31} {0, 75, 157, 118}

{155, 117, 138} {6, 133, 183, 172} {188, 27, 146, 41} {201, 174, 23, 64}

References [1] R.J.R. Abel, Existence of five MOLS of orders 18 and 60, J. Combin. Des. 23 (4) (2015) 135–139. [2] R.J.R. Abel, A.M. Assaf, I. Bluskov, M. Greig, N. Shalaby, New results on GDDs, covering, packing and directable designs with block size 5, J. Combin. Des. 18 (5) (2010) 337–368. [3] R.J.R. Abel, G. Ge, M. Greig, A.C.H. Ling, Further results on (v, {5, w ∗ }, 1)-PBDs, Discrete Math. 309 (8) (2009) 2323–2339. [4] F.E. Bennett, Y. Chang, G. Ge, M. Greig, Existence of (v, {5, w ∗ }, 1)-PBDs, Discrete Math. 279 (1–3) (2004) 61–105. In honour of Zhu Lie.

H. Wei, G. Ge / Discrete Mathematics 339 (2016) 1069–1085

1085

[5] C.J. Colbourn, J.H. Dinitz (Eds.), Handbook of Combinatorial Designs, second ed., in: Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2007. [6] C.J. Colbourn, D.R. Stinson, L. Zhu, More frames with block size four, J. Combin. Math. Combin. Comput. 23 (1997) 3–19. [7] P. Danziger, Uniform restricted resolvable designs with r = 3, Ars Combin. 46 (1997) 161–176. [8] J.H. Dinitz, A.C.H. Ling, P. Danziger, Maximum uniformly resolvable designs with block sizes 2 and 4, Discrete Math. 309 (14) (2009) 4716–4721. [9] J. Du, R.J.R. Abel, J. Wang, Some new resolvable gdds with k = 4 and doubly resolvable gdds with k = 3, Discrete Math. 338 (11) (2015) 2105–2118. [10] G. Ge, Uniform frames with block size four and index one or three, J. Combin. Des. 9 (1) (2001) 28–39. [11] G. Ge, Resolvable group divisible designs with block size four, Discrete Math. 243 (1–3) (2002) 109–119. [12] G. Ge, C.W.H. Lam, Resolvable group divisible designs with block size four and group size six, Discrete Math. 268 (1–3) (2003) 139–151. [13] G. Ge, C.W.H. Lam, A.C.H. Ling, Some new uniform frames with block size four and index one or three, J. Combin. Des. 12 (2) (2004) 112–122. [14] G. Ge, C.W.H. Lam, A.C.H. Ling, H. Shen, Resolvable maximum packings with quadruples, Des. Codes Cryptogr. 35 (3) (2005) 287–302. [15] G. Ge, A.C.H. Ling, A survey on resolvable group divisible designs with block size four, Discrete Math. 279 (1–3) (2004) 225–245. In honour of Zhu Lie. [16] G. Ge, A.C.H. Ling, Some more 5-GDDs and optimal (v, 5, 1)-packings, J. Combin. Des. 12 (2) (2004) 132–141. [17] G. Ge, A.C.H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs, J. Combin. Des. 13 (3) (2005) 222–237. [18] E.R. Lamken, W.H. Mills, R.S. Rees, Resolvable minimum coverings with quadruples, J. Combin. Des. 6 (6) (1998) 431–450. [19] R.S. Rees, Uniformly resolvable pairwise balanced designs with block sizes two and three, J. Combin. Theory Ser. A 45 (2) (1987) 207–225. [20] R.S. Rees, Two new direct product-type constructions for resolvable group-divisible designs, J. Combin. Des. 1 (1) (1993) 15–26. [21] R.S. Rees, Group-divisible designs with block size k having k + 1 groups, for k = 4, 5, J. Combin. Des. 8 (5) (2000) 363–386. [22] R.S. Rees, D.R. Stinson, Frames with block size four, Canad. J. Math. 44 (5) (1992) 1030–1049. [23] E. Schuster, Uniformly resolvable designs with index one and block sizes three and four—with three or five parallel classes of block size four, Discrete Math. 309 (8) (2009) 2452–2465. [24] E. Schuster, Small uniformly resolvable designs for block sizes 3 and 4, J. Combin. Des. 21 (11) (2013) 481–523. [25] E. Schuster, G. Ge, On uniformly resolvable designs with block sizes 3 and 4, Des. Codes Cryptogr. 57 (1) (2010) 45–69. With supplementary material available online. [26] H. Shen, J. Shen, Existence of resolvable group divisible designs with block size four. I, Discrete Math. 254 (1–3) (2002) 513–525. [27] H. Wei, G. Ge, Some more 5-GDDs, 4-frames and 4-RGDDs, Discrete Math. 336 (2014) 7–21. [28] J. Yin, A.C.H. Ling, C.J. Colbourn, R.J.R. Abel, The existence of uniform 5-GDDs, J. Combin. Des. 5 (4) (1997) 275–299.