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Constructions and Weight Distributions of Optimal Locally Repairable Codes.
- Source :
-
IEEE Transactions on Communications . May2022, Vol. 70 Issue 5, p2895-2908. 14p. - Publication Year :
- 2022
-
Abstract
- Locally repairable codes (LRCs) are important for distributed storage systems due to their efficient repairing ability of the failed storage nodes. A $q$ -ary optimal $(n,k,r)$ -LRC is an $[n,k,d]$ linear code over $\mathbb {F}_{q}$ such that every code symbol has locality $r$ , and the minimum distance attains the well-known Singleton-like bound. In this paper, we study the maximal code length, code constructions and weight distributions of $q$ -ary optimal LRCs with locality 2 and distance 5, which are of both practical and theoretical interest. Firstly, it is proved that when the code dimension is even or odd, corresponding maximal code lengths of such $q$ -ary optimal LRCs are $3 \cdot \lfloor \frac {q+1}{3} \rfloor $ and $3 \cdot \left \lfloor{ \frac {q-1}{3} }\right \rfloor +5$ , respectively. Up to the equivalence of linear codes, we propose constructions of all the possible $q$ -ary optimal LRCs with locality 2, distance 5 and maximal code length. Then, by characterizing the weight type hierarchy of codewords, we show that the weight distribution of any $q$ -ary optimal LRC with locality 2, distance 5 and even code dimension can be uniquely determined and explicit expression of the weight distribution is given. Moreover, it is shown that all $q$ -ary optimal LRCs with locality 2, distance 5 and even code dimension are maximally recoverable. [ABSTRACT FROM AUTHOR]
- Subjects :
- *LINEAR codes
*DISMISSAL of employees
Subjects
Details
- Language :
- English
- ISSN :
- 00906778
- Volume :
- 70
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Communications
- Publication Type :
- Academic Journal
- Accession number :
- 156931617
- Full Text :
- https://doi.org/10.1109/TCOMM.2022.3155165