Optimal binary and ternary locally repairable codes with minimum distance 6

A locally repairable code (LRC) is a code that can recover any symbol of a codeword by reading at most r other symbols, denoted by r -LRC. In this paper, we study binary and ternary linear LRCs with disjoint repair groups and minimum distance d = 6. Using the intersection subspaces technique, we exp...

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Bibliographic Details
Published in:Designs, codes, and cryptography codes, and cryptography, 2024-05, Vol.92 (5), p.1251-1265
Main Authors: Zhang, Wenqin, Luo, Yuan, Wang, Lele
Format: Article
Language:English
Subjects:
Codes
Coding and Information Theory
Computer Science
Cryptology
Discrete Mathematics in Computer Science
Parameters
Subspaces
Upper bounds
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Summary:A locally repairable code (LRC) is a code that can recover any symbol of a codeword by reading at most r other symbols, denoted by r -LRC. In this paper, we study binary and ternary linear LRCs with disjoint repair groups and minimum distance d = 6. Using the intersection subspaces technique, we explicitly construct dimensional optimal LRCs. First, based on the intersection subspaces constructed by t -spread, a construction of binary LRCs is designed. Particularly, a class of binary linear LRCs with r = 11 is optimal in terms of achieving a sphere-packing type upper bound. Next, by using the Kronecker product of two matrices, two classes of dimensional optimal ternary LRCs with small locality ( r = 3, 5) are presented. Compared to previous results, our construction is more flexible regarding code parameters. Finally, we also discuss the parameters of a code obtained by applying a shortening operation to our LRCs. We show that these shortened LRCs are also k -optimal.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-023-01341-2