Constructions and Weight Distributions of Optimal Locally Repairable Codes

• Published in: IEEE transactions on communications Vol. 70; no. 5; pp. 2895 - 2908
• Main Authors: Hao, Jie, Zhang, Jun, Xia, Shu-Tao, Fu, Fang-Wei, Yang, Yixian
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Codes; Distributed databases; Distributed storage systems; Fault tolerant systems; Linear codes; locally repairable codes; Maintenance engineering; maximal recoverability; Singleton-like bound; Storage systems; Termination of employment; Upper bound; weight distribution
• ISSN: 0090-6778; 1558-0857
• DOI: 10.1109/TCOMM.2022.3155165

Locally repairable codes (LRCs) are important for distributed storage systems due to their efficient repairing ability of the failed storage nodes. A <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-ary optimal <inline-formula> <tex-math notation="LaTeX">(n,k,r) </tex-math></inline-formula>-LRC is an <inline-formula> <tex-math notation="LaTeX">[n,k,d] </tex-math></inline-formula> linear code over <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{q} </tex-math></inline-formula> such that every code symbol has locality <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-ary optimal LRCs are <inline-formula> <tex-math notation="LaTeX">3 \cdot \lfloor \frac {q+1}{3} \rfloor </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">3 \cdot \left \lfloor{ \frac {q-1}{3} }\right \rfloor +5 </tex-math></inline-formula>, respectively. Up to the equivalence of linear codes, we propose constructions of all the possible <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-ary optimal LRCs with locality 2, distance 5 and even code dimension are maximally recoverable.