Recovery Sets of Subspaces from a Simplex Code (2403.20170v1)

Abstract: Recovery sets for vectors and subspaces are important in the construction of distributed storage system codes. These concepts are also interesting in their own right. In this paper, we consider the following very basic recovery question: what is the maximum number of possible pairwise disjoint recovery sets for each recovered element? The recovered elements in this work are d-dimensional subspaces of a $k$-dimensional vector space over GF(q). Each server stores one representative for each distinct one-dimensional subspace of the k-dimensional vector space, or equivalently a distinct point of PG(k-1,q). As column vectors, the associated vectors of the stored one-dimensional subspaces form the generator matrix of the $[(q^k -1)/(q-1),k,q^{k-1}]$ simplex code over GF(q). Lower bounds and upper bounds on the maximum number of such recovery sets are provided. It is shown that generally, these bounds are either tight or very close to being tight.