A Class of $(n, k, r, t)_i$ LRCs Via Parity Check Matrix

Deep Mukhopadhyay; Sanjit Bhowmick; Kalyan Hansda; Satya Bagchi

A Class of $(n, k, r, t)_i$ LRCs Via Parity Check Matrix

Information Theory 2022-08-25 v2 Discrete Mathematics math.IT

Authors: Deep Mukhopadhyay , Sanjit Bhowmick , Kalyan Hansda , Satya Bagchi

Abstract

A code is called $(n, k, r, t)$ information symbol locally repairable code \big( $(n, k, r, t)_i$ LRC\big) if each information coordinate can be achieved by at least $t$ disjoint repair sets, containing at most $r$ other coordinates. This paper considers a class of $(n, k, r, t)_i$ LRCs, where each repair set contains exactly one parity coordinate. We explore the systematic code in terms of the standard parity check matrix. First, some structural features of the parity check matrix are proposed by showing some connections with the membership matrix and the minimum distance optimality of the code. Next to that, parity check matrix based proofs of various bounds associated with the code are placed. In addition to this, we provide several constructions of optimal $(n, k, r, t)_i$ LRCs, with the help of two Cayley tables of a finite field. Finally, we generalize a result of $q$ -ary $(n, k, r)$ LRCs to $q$ -ary $(n, k, r, t)$ LRCs.

Keywords

locally repairable codes information symbol parity check matrix minimum distance optimal constructions finite field cayley table

Cite

@article{arxiv.2112.05474,
  title  = {A Class of $(n, k, r, t)_i$ LRCs Via Parity Check Matrix},
  author = {Deep Mukhopadhyay and Sanjit Bhowmick and Kalyan Hansda and Satya Bagchi},
  journal= {arXiv preprint arXiv:2112.05474},
  year   = {2022}
}

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12 pages

A Class of $(n, k, r, t)_i$ LRCs Via Parity Check Matrix

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Keywords

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