[Paper Review] Constructions of Optimal Cyclic $(r,δ)$ Locally Repairable Codes

Bin Chen, Shu‐Tao Xia+2 more|arXiv (Cornell University)|Sep 5, 2016

Advanced Data Storage Technologies23 references3 citations

TL;DR

This paper constructs optimal $q$-ary cyclic $(r,\delta)$ locally repairable codes (LRCs) for lengths dividing $q-1$ and $q+1$, generalizing prior work on $r$-local LRCs. By leveraging the structure of zeros in Reed-Solomon codes and Berlekamp-Justesen codes, the authors present new constructions that achieve the Singleton-like bound, with explicit parameter sets for $n \mid q+1$ and $\delta \geq 2$, including codes of length $q+1$ with the best-known parameters for $d > 4$. The key contribution is a systematic method to build optimal cyclic LRCs beyond the $n \mid q-1$ case, expanding the known parameter space for efficient distributed storage.

ABSTRACT

A code is said to be a $r$-local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most $r$ other coordinates. When some of the $r$ coordinates are also erased, the $r$-local LRC can not accomplish the local repair, which leads to the concept of $(r,δ)$-locality. A $q$-ary $[n, k]$ linear code $\cC$ is said to have $(r, δ)$-locality ($δ\ge 2$) if for each coordinate $i$, there exists a punctured subcode of $\cC$ with support containing $i$, whose length is at most $r + δ- 1$, and whose minimum distance is at least $δ$. The $(r, δ)$-LRC can tolerate $δ-1$ erasures in total, which degenerates to a $r$-local LRC when $δ=2$. A $q$-ary $(r,δ)$ LRC is called optimal if it meets the Singleton-like bound for $(r,δ)$-LRCs. A class of optimal $q$-ary cyclic $r$-local LRCs with lengths $n\mid q-1$ were constructed by Tamo, Barg, Goparaju and Calderbank based on the $q$-ary Reed-Solomon codes. In this paper, we construct a class of optimal $q$-ary cyclic $(r,δ)$-LRCs ($δ\ge 2$) with length $n\mid q-1$, which generalizes the results of Tamo \emph{et al.} Moreover, we construct a new class of optimal $q$-ary cyclic $r$-local LRCs with lengths $n\mid q+1$ and a new class of optimal $q$-ary cyclic $(r,δ)$-LRCs ($δ\ge 2$) with lengths $n\mid q+1$. The constructed optimal LRCs with length $n=q+1$ have the best-known length $q+1$ for the given finite field with size $q$ when the minimum distance is larger than $4$.

Motivation & Objective

• To extend the construction of optimal $r$-local LRCs to the more general $(r,\delta)$-locality model with $\delta \geq 2$.
• To develop new constructions of optimal $q$-ary cyclic LRCs for code lengths dividing $q+1$, which were previously unexplored in optimal cyclic settings.
• To achieve the generalized Singleton-like bound for $(r,\delta)$-LRCs using cyclic code structures over finite fields.
• To provide explicit constructions of optimal cyclic LRCs with length $n = q+1$, which achieve the best-known length for given $q$ when $d > 4$.
• To explore the existence and structure of optimal cyclic LRCs beyond the $n \mid q-1$ case, addressing a gap in the literature.

Proposed method

• Generalize the zero-set structure of Reed-Solomon codes used by Tamo et al. to construct optimal cyclic $(r,\delta)$-LRCs for $n \mid q-1$.
• Use the defining set of zeros of cyclic codes, specifically selecting sets $D$ and $L_m$ to ensure both locality and minimum distance constraints.
• Construct cyclic codes with defining sets $D \cup (L_1 \cup L_{-1})$ where $L_m$ are arithmetic progressions modulo $n$, ensuring $(r,\delta)$-locality.
• Apply the Berlekamp-Justesen code framework to construct new optimal $q$-ary cyclic $r$-local and $(r,\delta)$-LRCs for $n \mid q+1$.
• Employ algebraic number theory and properties of primitive $n$th roots of unity in $\mathbb{F}_{q^2}$ to ensure the code is cyclic and meets the Singleton-like bound.
• Use parameter-dependent zero sets $D$ based on $k$, $r$, $\delta$, $\mu$, and parity of $q$, $\mu$, and $\nu'$ to control dimension and locality.

Experimental results

Research questions

• RQ1 Can optimal cyclic $(r,\delta)$-LRCs be constructed for lengths dividing $q+1$, beyond the known $n \mid q-1$ case?
• RQ2 What structural conditions on the defining set of zeros ensure both optimal distance and $(r,\delta)$-locality in cyclic codes?
• RQ3 Do optimal cyclic $(r,\delta)$-LRCs with length $n = q+1$ exist for $\delta \geq 2$ and $d > 4$, and what are their parameters?
• RQ4 How can the Berlekamp-Justesen code construction be adapted to yield optimal cyclic LRCs with $n \mid q+1$?
• RQ5 What is the maximal possible length $n$ of a $q$-ary optimal cyclic $(r,\delta)$-LRC for given $q$, $k$, $r$, and $\delta$?

Key findings

• The authors construct a new class of optimal $q$-ary cyclic $(r,\delta)$-LRCs with length $n \mid q+1$, extending known results that were limited to $n \mid q-1$.
• For $n = q+1$, the constructed codes achieve the best-known length for a given finite field size $q$ when the minimum distance exceeds 4.
• The constructions for $n \mid q+1$ are based on Berlekamp-Justesen codes and involve carefully chosen defining sets of zeros with arithmetic progression structure.
• The paper generalizes Tamo et al.'s $r$-local LRCs to $(r,\delta)$-locality, achieving the generalized Singleton-like bound for $\delta \geq 2$.
• Explicit constructions are provided for multiple cases based on the parity of $q$, $\mu$, and $\nu'$, with corresponding zero sets $D$ and locality sets $L_m$.
• The results show that optimal cyclic $(r,\delta)$-LRCs exist for $n \mid q+1$ under specific parameter conditions, as detailed in Table III.