Rate optimal binary linear locally repairable codes with small availability

Conference Paper

A locally repairable code with availability has the property that every code symbol can be recovered from multiple, disjoint subsets of other symbols of small size. In particular, a code symbol is said to have (r, t)-availability if it can be recovered from t disjoint subsets, each of size at most r. A code with availability is said to be rate optimal, if its rate is maximum among the class of codes with given locality, availability, and alphabet size. This paper focuses on rate-optimal binary, linear codes with small availability, and makes three contributions. First, it establishes tight upper bounds on the rate of binary linear codes with (r, 2) and (2, 3) availability. Second, it establishes a uniqueness result for binary rate-optimal codes, showing that for certain classes of binary linear codes with (r, 2) and (2, 3)-availability, any rate-optimal code must be a direct sum of shorter rateoptimal codes. Finally, it presents a class of locally repairable codes associated with convex polyhedra, especially, focusing on the codes associated with the Platonic solids. It demonstrates that these codes are locally repairable with t = 2, and that the codes associated with (geometric) dual polyhedra are (coding theoretic) duals of each other.

Full Text

Duke Authors

Cited Authors

  • Kadhe, S; Calderbank, R

Published Date

  • August 9, 2017

Published In

Start / End Page

  • 166 - 170

International Standard Serial Number (ISSN)

  • 2157-8095

International Standard Book Number 13 (ISBN-13)

  • 9781509040964

Digital Object Identifier (DOI)

  • 10.1109/ISIT.2017.8006511

Citation Source

  • Scopus