New Constructions of SD and MR Codes over Small Finite Fields

TL;DR

This paper introduces new explicit constructions of SD and MR erasure-correcting codes over small finite fields, which are crucial for efficient data storage and recovery.

Contribution

The paper provides novel explicit constructions of SD and MR codes over small finite fields, improving practical applicability in data storage systems.

Findings

Constructed SD and MR codes over small finite fields.

Codes correct all relevant erasure patterns.

Enhanced efficiency in encoding and decoding processes.

Abstract

Data storage applications require erasure-correcting codes with prescribed sets of dependencies between data symbols and redundant symbols. The most common arrangement is to have $k$ data symbols and $h$ redundant symbols (that each depends on all data symbols) be partitioned into a number of disjoint groups, where for each group one allocates an additional (local) redundant symbol storing the parity of all symbols in the group. A code as above is maximally recoverable, if it corrects all erasure patterns that are information theoretically correctable given the dependency constraints. A slightly weaker guarantee is provided by SD codes. One key consideration in the design of MR and SD codes is the size of the finite field underlying the code as using small finite fields facilitates encoding and decoding operations. In this paper we present new explicit constructions of SD and MR codes over small finite fields.