Recovery Sets of Subspaces from a Simplex Code
Yeow Meng Chee, Tuvi Etzion, Han Mao Kiah, Hui Zhang
TL;DR
This work analyzes the maximum number $N_q(k,d)$ of pairwise disjoint recovery sets that can recover any given $d$-subspace of $\\mathbb{F}_q^k$ when servers store all $1$-subspaces (points of $PG(k-1,q)$) corresponding to the simplex-code. It develops both lower and upper bounds using a spectrum of methods: binary-perfect-code constructions, integer-programming formulations, and rank-metric (MRD) code-based partitions, extended with projective-geometry partitions for general $q$. The paper provides exact results for several low-dim binary cases (notably $d=2$) and tight or near-tight bounds for larger $d$ and for certain $q>2$, along with explicit constructions (quintriples, MRD lifts) that realize many recovery sets. The methods illuminate how to design and analyze availability and recovery in subspace-based distributed storage codes and have implications for network coding and storage systems where subspace data must be reliably recovered from disjoint server groups.
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.