On subspace designs

The aim of this paper is to investigate the theory of subspace designs, which have been originally introduced by Guruswami and Xing in 2013 to give the first construction of positive rate rank-metric codes list decodable beyond half the distance. However, sets of subspaces with special pattern of intersections with other subspaces have been already studied, such as spreads, generalised arcs and caps, and Cameron-Liebler sets. In this paper we provide bounds involving the dimension of the subspaces forming a subspace design and the parameters of the ambient space, showing constructions satisfying the equality in such bounds. Then we also introduce two dualities relations among them. Among subspace designs, those that we call s-designs are central in this paper, as they generalize the notion of s-scattered subspace to subspace design and for their special properties. Indeed, we prove that, for certain values of s, they correspond to the optimal subspace designs, that is those subspace designs that are associated with linear maximum sum-rank metric codes. Special attention has been paid for the case s=1 for which we provide several examples, yielding surprising to families of two-intersection sets with respect to hyperplanes (and hence two-weight linear codes). Moreover, s-designs can be used to construct explicit lossless dimension expanders (a linear-algebraic analogue of expander graphs), without any restriction on the order of the field. Another class of subspace designs we study is those of cutting designs, since they extend the notion of cutting blocking set recently introduced by Bonini and Borello. These designs turn out to be very interesting as they in one-to-one correspondence with minimal sum-rank metric codes. The latter codes have been introduced in this paper and they naturally extend the notions of minimal codes in both Hamming and rank metrics.