Singleton-Type Optimal LRCs with Minimum Distance 3 and 4 from Projective Code

Qiang FU
Ruihu LI
Luobin GUO

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E104-A    No.1    pp.319-323
Publication Date: 2021/01/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.2019EAL2158
Type of Manuscript: LETTER
Category: Coding Theory
Singleton-type bound,  optimal locally repairable codes,  projective cap,  

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Locally repairable codes (LRCs) are implemented in distributed storage systems (DSSs) due to their low repair overhead. The locality of an LRC is the number of nodes in DSSs that participate in the repair of failed nodes, which characterizes the repair cost. An LRC is called optimal if its minimum distance attains the Singleton-type upper bound [1]. In this letter, optimal LRCs are considered. Using the concept of projective code in projective space PG(k, q) and shortening strategy, LRCs with d=3 are proposed. Meantime, derived from an ovoid [q2+1, 4, q2]q code (responding to a maximal (q2+1)-cap in PG(3, q)), optimal LRCs over Fq with d=4 are constructed.