Optimal (<I>r</I>, δ)-Locally Repairable Codes from Reed-Solomon Codes

Lin-Zhi SHEN; Yu-Jie WANG

doi:10.1587/transfun.2023EAL2026

Optimal (r, δ)-Locally Repairable Codes from Reed-Solomon Codes

Lin-Zhi SHEN, Yu-Jie WANG

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For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs), its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in F_q. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over F_q provided that r + δ + s - 1≤q, s≤δ, s<r, and any positive t. The code length is unbounded. The second class gives the [r + r' + d + δ - 2, r + r', d] optimal (r, δ)-LRC over F_q provided that r - r'≥d - δ and r + d - 1≤q + 1, which will produce optimal (r, δ)-LRCs with large minimum distance.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E106-A No.12 pp.1589-1592

Publication Date: 2023/12/01

Publicized: 2023/05/30

Online ISSN: 1745-1337

DOI: 10.1587/transfun.2023EAL2026

Type of Manuscript: LETTER

Category: Coding Theory

Authors

Lin-Zhi SHEN
Civil Aviation University of China
Yu-Jie WANG
Civil Aviation University of China

Keyword

locally repairable code, (r, δ)-locally repairable code, Reed-Solomon code, Singleton-like bound

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Lin-Zhi SHEN, Yu-Jie WANG, "Optimal (r, δ)-Locally Repairable Codes from Reed-Solomon Codes" in IEICE TRANSACTIONS on Fundamentals, vol. E106-A, no. 12, pp. 1589-1592, December 2023, doi: 10.1587/transfun.2023EAL2026.
Abstract: For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs), its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in F_q. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over F_q provided that r + δ + s - 1≤q, s≤δ, s<r, and any positive t. The code length is unbounded. The second class gives the [r + r' + d + δ - 2, r + r', d] optimal (r, δ)-LRC over F_q provided that r - r'≥d - δ and r + d - 1≤q + 1, which will produce optimal (r, δ)-LRCs with large minimum distance.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2023EAL2026/_p

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@ARTICLE{e106-a_12_1589,
author={Lin-Zhi SHEN, Yu-Jie WANG, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Optimal (r, δ)-Locally Repairable Codes from Reed-Solomon Codes},
year={2023},
volume={E106-A},
number={12},
pages={1589-1592},
abstract={For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs), its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in F_q. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over F_q provided that r + δ + s - 1≤q, s≤δ, s<r, and any positive t. The code length is unbounded. The second class gives the [r + r' + d + δ - 2, r + r', d] optimal (r, δ)-LRC over F_q provided that r - r'≥d - δ and r + d - 1≤q + 1, which will produce optimal (r, δ)-LRCs with large minimum distance.},
keywords={},
doi={10.1587/transfun.2023EAL2026},
ISSN={1745-1337},
month={December},}

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TY - JOUR
TI - Optimal (r, δ)-Locally Repairable Codes from Reed-Solomon Codes
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1589
EP - 1592
AU - Lin-Zhi SHEN
AU - Yu-Jie WANG
PY - 2023
DO - 10.1587/transfun.2023EAL2026
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E106-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2023
AB - For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs), its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in F_q. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over F_q provided that r + δ + s - 1≤q, s≤δ, s<r, and any positive t. The code length is unbounded. The second class gives the [r + r' + d + δ - 2, r + r', d] optimal (r, δ)-LRC over F_q provided that r - r'≥d - δ and r + d - 1≤q + 1, which will produce optimal (r, δ)-LRCs with large minimum distance.
ER -