Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes

TL;DR

This paper studies optimal cyclic locally recoverable codes (LRCs), characterizes their zeros, explores subfield subcodes, and establishes bounds on their dual distance and minimum distance.

Contribution

It provides a characterization of optimal cyclic LRC codes via their zeros and analyzes subfield subcodes, offering new bounds on their parameters.

Findings

Characterization of optimal cyclic LRC codes through zeros

Analysis of subfield subcodes and their locality

Upper bounds on dual distance and minimum distance

Abstract

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalize the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper we focus on optimal cyclic codes that arise from this 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. The locality parameter of a cyclic code is related to the dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.