A new series of large sets of subspace designs over the binary field
Michael Kiermaier, Reinhard Laue, Alfred Wassermann
TL;DR
This work advances the theory of subspace designs by constructing a new large set $LS_2[3](2,4,8)$ computationally and embedding it in a recursive framework to produce an infinite family of large sets $LS_2[3](2,k,v)$ for $v\ge 8$ under a modular condition on $v$ and $k$. The authors develop and apply the Kramer–Mesner method with a group action to obtain $G$-invariant designs, and leverage a Graßmannian join decomposition to extend these base cases into an entire infinite series. The main result, proven by induction on $v$, shows existence for all $v,k$ with $2\le \bar{v} < \bar{k} \le 5$, conditioned on the base cases $LS_2[3](2,3,8)$, $LS_2[3](2,4,8)$ and their duals. This contributes concrete, constructive tools and broadens the landscape of $q$-analogs of combinatorial design theory, with potential implications for parallelisms and network coding over $GF(2)$.
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 Graßmannian into joins yields a construction for the claimed large sets.