Constacyclic locally recoverable codes from their duals

Abstract

A code has locality r if a symbol in any coordinate of a codeword in the code can be recovered by accessing the value of at most r other coordinates. Such codes are called locally recoverable codes (LRCs for short). Since LRCs can recover a failed node by accessing the minimum number of the surviving nodes, these codes are used in distributed storage systems such as Microsoft Azure. In this paper, constacyclic LRCs are obtained from their parity-check polynomials. Constacyclic codes with locality r≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\le 2$$\\end{document} and dimension 2 are obtained and a sufficient and necessary condition for these codes to have locality 1 is given. Then, the construction is generalized. Distance optimal constacyclic LRCs with distance 2 are obtained. Also, constacyclic codes with locality 1 are constructed. They may be so useful in practice thanks to their minimum locality. Constacyclic codes whose locality is equal to their dimension are given. Furthermore, constacyclic LRCs are obtained from cyclotomic cosets.