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

TL;DR

This paper proves the existence of large sets of subspace designs over the binary field for infinitely many parameters, using a combination of computer-aided construction and recursive combinatorial methods.

Contribution

It introduces a new family of large sets of subspace designs over GF(2), with explicit constructions and conditions for their existence.

Findings

Constructed a computer-based example of an LS_2[3](2,4,8) design.

Established existence conditions for large sets over binary fields for various parameters.

Applied recursive methods to extend the constructions to infinitely many cases.

Abstract

In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2 \leq \bar{v} < \bar{k} \leq 5$. The proof is constructive and consists of two parts. First, we give a computer construction for an $\operatorname{LS}_2[3](2,4,8)$, 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 $\operatorname{LS}_2[3](2,3,8)$, the application of a recursion method based on a decomposition of the Gra{\ss}mannian into joins yields a construction for the claimed large sets.