A new series of large sets of subspace designs over the binary field

Abstract

In this article, we show the existence of large sets LS 2 [3](2, k, v) for infinitely many values of k and v. The exact condition is v ≥ 8 and 0 ≤ k ≤ v such that for the remaindersv andk of v and k modulo 6 we have 2 ≤v < k ≤ 5.The proof is constructive and consists of two parts. First, we give a computer construction for an LS 2 [3](2, 4, 8), which is a partition of the set of all 4-dimensional subspaces of an 8-dimensional vector space over the binary field into three disjoint 2-(8, 4, 217) 2 subspace designs. Together with the already known LS 2 [3] (2,3,8), the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets.

New 2-designs over finite fields from derived and residual designs

“…As an immediate consequence of this theorem for t = 3 and q = 2 applied to Table 1 containing 2-(9, k, λ; 2) designs for k ∈ {3, 4} we obtain the following corollary: 7,12,19,21,22,24,31,36,42,43,48,49,55, 60, 63 designs with parameters 2-(9, 3, λ; 2) and 2-(9, 4, 21λ; 2) do exist which imply the existence of further designs with the following parameters:…”

“…Since the first non-trivial t-(n, k, λ; q) designs for t > 1 were introduced in 1987 [18] the interest in these objects has increased. Several results on parameter sets of new constructed designs over finite fields have been published [2,3,4,5,6,7,8,9,10,11,12,14,15,17], whereas until now only two infinite series for arbitrary field size q are known:…”

Subspace packings: constructions and bounds

“…for each 1 ≤ j ≤ s. Theorem 5 can be applied for a limited number of parameters. The best are based on partitioning of all k-subspaces into such designs as discussed in [12,13,54,59]. There are other with smaller t, especially when t = 1.…”