The completion determination of optimal $$(3,4)$$ ( 3 , 4 ) -packings

Abstract

A 3-(n, 4, 1) packing design consists of an n-element set X and a collection of 4-element subsets of X, called blocks, such that every 3-element subset of X is contained in at most one block. The packing number of quadruples d(3, 4, n) denotes the number of blocks in a maximum 3-(n, 4, 1) packing design, which is also the maximum number A(n, 4, 4) of codewords in a code of length n, constant weight 4, and minimum Hamming distance 4. In this paper the undecided 21 packing numbers A(n, 4, 4) are shown to be equal to Johnson bound J(n,4,4) 47, 75, 77, 79, 159}.

Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes

“…Let n = (n), then a 3-(n, k, 1) generalized packing design is indeed a 3-(n, 4, 1) packing, for which the determination of packing numbers D(n, 4, 3) has been completed by Bao and Ji in [19]. Hence, we have the following result.…”

Suitable sets of permutations, packings of triples, and Ramsey’s theorem

“…Given t, k, and l, the determination of the packing number D(l, k, t), the maximum size of a t-(l, k, 1) packing, constitutes a central problem in combinatorial design theory, as well as in coding theory [13]. When k = 4 and t = 3, the value of D(l, 4, 3) has been completely determined by constructive methods, see [14,15,16,17], and it achieves the well known Johnson bound given below:…”

Combinatorial constructions of optimal (m, n, 4, 2) optical orthogonal signature pattern codes

“…{(0, 0), (0, 1), (0, 11), (0, 6)}, {(0, 0), (1, 1), (2,11), (0, 6)}, {(0, 0), (1,3), (2,9), (0, 6)}, {(0, 0), (1,5), (2,7), (0, 6)}, {(0, 0), (0, 1), (0, 3), (0, 4)}, {(0, 0), (0, 1), (0, 5), (0, 8)}, {(0, 0), (0, 1), (1, 0), (1, 1)}, {(0, 0), (0, 1), (1,2), (1, 3)}, {(0, 0), (0, 1), (1,4), (1,5)}, {(0, 0), (0, 1), (1,6), (1, 7)}, {(0, 0), (0, 1), (1,8), (1,9)}, {(0, 0), (0, 1), (1,10), (1,11)}, {(0, 0), (0, 2), (0, 5), (1, 0)}, {(0, 0), (0, 10), (0, 7), (2, 0)}, {(0, 0), (0, 2), (0, 6), (1, 2)}, {(0, 0), (0, 10), (0, 6), (2, 10)}, {(0, 0), (0, 2), (1, 1), (1, 3)}, {(0, 0), (0, 2), (1,4), (1,6)}, {(0, 0), (0, 2), (1,5), (1,7)}, {(0, 0), (0, 2), (1,8), (1,10)}, {(0, 0), (0, 2), (1,9), (1,11) Proof Start with a strictly Z m × Z 2 ǫ ·3n -invariant G * (m, 2 ε · 3n, 4, 3) design relative to {0} × Z 2 ǫ ·3n , ǫ ∈ {1, 2}, which exists from the proof of Theorem 6.5. Applying Construction 4.2 gives a strictly Z 3m × Z 2 ǫ ·3n -invariant G * (m, 2 ε · 9n, 4, 3) design relative to mZ 3m × Z 2 ǫ ·3n .…”

“…Construct a strictly semi-cyclic G(2,8,4, 3) design on {0, 1} × Z 8 with groups {i} × Z 8 , i ∈ {0, 1}, whose set F of base blocks consists of the following:{(0, 0), (0, 1), (1, 0), (1, 1)}, {(0, 0), (0, 1),(1,2),(1,4)}, {(0, 0), (0, 1),(1,3), (1, 7)}, {(0, 0), (0, 1),(1,5),(1,6)}, {(0, 0), (0, 2), (1, 0), (1, 2)}, {(0, 0), (0, 2), (1, 1),(1,6)}, {(0, 0), (0, 2), (1, 3), (1, 4)}, {(0, 0), (0, 2), (1, 5), (1, 7)}, {(0, 0), (0, 3), (1, 0), (1, 4)}, {(0, 0), (0, 3), (1, 1), (1, 7)}, {(0, 0), (0, 3), (1, 2), (1, 5)}, {(0, 0), (0, 3), (1, 3), (1, 6)}, {(0, 0), (0, 4), (1, 0), (1, 5)}, {(0, 0), (0, 4), (1, 2), (1, 3)}.Then D ∪ F is the set of base blocks of a strictlyZ 4 × Z 8 -invariant G(2, 16, 4, 3) design with groups {i, i + 2} × Z 8 , 0 ≤ i < 2.Since there is a (2, 8, 4, 2)-OOSPC with J(2, 8, 4, 2) codewords from[45], there is a strictly Z 2 ×Z 8invariant P QS(16)with J(2, 8, 4, 2) base blocks by Theorem 2.1. By Construction 3.3, there is a strictly Z 4 ×Z 8 -invariant P QS(32) with J(4, 8, 4, 2) base blocks, which leads to an optimal (4, 8, 4, 2)-OOSPC with the size meeting the upper bound (1.1) by Theorem 2.1.…”