Sparse and Balanced MDS Codes over Small Fields

TL;DR

This paper demonstrates the existence of sparse and balanced MDS codes over small fields for certain parameters, and provides polynomial-time algorithms for their construction, enhancing distributed storage efficiency.

Contribution

It proves the existence of such codes over small fields for n ≤ 2k and develops polynomial-time algorithms for their construction.

Findings

Existence of sparse, balanced MDS codes over small fields for n ≤ 2k

Polynomial-time algorithms for code construction

Applicable to distributed storage systems

Abstract

Maximum Distance Separable (MDS) codes with a sparse and balanced generator matrix are appealing in distributed storage systems for balancing and minimizing the computational load. Such codes have been constructed via Reed-Solomon codes over large fields. In this paper, we focus on small fields. We prove that there exists an $[n,k]_q$ MDS code that has a sparse and balanced generator matrix for any $q\geq n$ provided that $n\leq 2k$, by designing several algorithms with complexity running in polynomial time in $k$ and $n$.