Haoming Shi

Yuan Gao, Haoming Shi, Weijun Fang

Constructing optimal $(r,δ)$-LRCs that attain the Singleton-type bound is an active and important research direction, particularly due to their practical applications in distributed storage systems. In this paper, we focus on the construction of optimal $(r,δ)$-LRCs with flexible minimum distances, especially for the case $δ\geq 3$. We first extend a general framework -- originally proposed by Li \textit{et al.} (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) -- for constructing optimal $r$-LRCs via automorphism groups of elliptic function fields to the case of $(r,δ)$-LRCs. This newly extended general framework relies on certain conditions concerning the group law of elliptic curves. By carefully selecting elliptic function fields suitable for this framework, we arrive at several families of explicit $q$-ary optimal $(r,3)$-LRCs and $(2,δ)$-LRCs with lengths slightly less than $q + 2\sqrt{q}$. Next, by employing automorphism groups of hyperelliptic function fields of genus $2$, we develop a framework for constructing optimal $(r,3)$-LRCs and obtain a family of explicit $q$-ary optimal $(4,3)$-LRCs with code lengths slightly below $q+4\sqrt{q}$. We then consider the construction of optimal $(r,δ)$-LRCs via hyperelliptic function fields of arbitrary genus $g \geq 2$, yielding a class of explicit $q$-ary optimal $(g+1-g',g+1+g')$-LRCs for $0 \leq g' \leq g-1$ with lengths up to $q + 2g\sqrt{q}$. Finally, applying certain superelliptic curves derived from modified Norm-Trace curves, we construct two families of explicit optimal $(r,δ)$-LRCs with even longer code lengths and more flexible parameters. Notably, many of the newly constructed optimal $(r,δ)$-LRCs attain the largest known lengths among existing constructions with flexible minimum distances.

Haoming Shi, Weijun Fang

Erasure codes are a key technique for achieving fault-tolerant storage in modern distributed storage systems. As storage systems evolve, their code parameters often need to be adjusted to accommodate changes in storage scale, reliability requirements, and disk failure rates. Such adaptation is performed through code conversion, where data encoded under an initial code are transformed into data encoded under a final code. Convertible codes are designed to carry out this transformation efficiently while preserving desirable properties of the underlying codes. In this work, we study conversions between systematic optimal-distance locally repairable codes (LRCs), focusing on the read-bandwidth cost in the global split regime. We concentrate on the parameter range \(g^I,g^F\le r\), where the numbers of initial and final global parity nodes do not exceed the local dimension. For this entire range, we derive read-bandwidth lower bounds for stable optimal-distance locally repairable convertible codes (LRCCs) via an information-theoretic approach, without imposing any linearity assumption on the initial and final codes or on the conversion procedure. We then develop constructions based on MDS array codes with prescribed repair properties. According to the relative sizes of \(g^I\) and \(g^F\), the construction is divided into the three cases \(g^F=g^I\), \(g^F>g^I\), and \(g^F<g^I\); in each case, it attains the corresponding lower bound. Hence we characterize the optimal read-bandwidth cost for stable optimal-distance LRCCs throughout the parameter range \(g^I,g^F\le r\).