Cyclic LRC Codes and their Subfield Subcodes

TL;DR

This paper studies optimal cyclic locally recoverable codes (LRCs), characterizes them via their zeros, and explores subfield subcodes, revealing multiple construction methods and their properties.

Contribution

It provides a zero-based characterization of optimal cyclic LRC codes and analyzes subfield subcodes, expanding understanding of their structure and locality.

Findings

Characterization of optimal cyclic LRC codes via zeros

Multiple equivalent constructions of these codes

Results on locality and minimum distance of subfield subcodes

Abstract

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalizes the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Transactions on Information Theory, no. 8, 2014; arXiv:1311.3284). In this paper we focus on the optimal cyclic codes that arise from the general construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance.