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A class of locally recoverable codes over finite chain rings

Giulia Cavicchioni, Eleonora Guerrini, Alessio Meneghetti

TL;DR

This work extends locally recoverable codes to finite chain rings, generalizing field-based results by developing a TB-type polynomial-interpolation framework over rings. It establishes a ring-aware LRC distance bound $d \le n - K - \lceil K/r\rceil + 2$ and constructs optimal codes using good polynomials, including over Galois rings, with explicit encoding and decoding schemes. The authors further generalize the construction to broader ring settings, remove length limitations via well- and non-well-conditioned sets, and connect maximum possible code lengths to residue-field bounds, offering a path toward longer LRCs in ring alphabets. The results demonstrate the practicality and versatility of ring-based TB constructions for robust distributed storage, while outlining directions for longer codes and richer polynomial families.

Abstract

Locally recoverable codes deal with the task of reconstructing a lost symbol by relying on a portion of the remaining coordinates smaller than an information set. We consider the case of codes over finite chain rings, generalizing known results and bounds for codes over fields. In particular, we propose a new family of locally recoverable codes by extending a construction proposed in 2014 by Tamo and Barg, and we discuss its optimality.

A class of locally recoverable codes over finite chain rings

TL;DR

This work extends locally recoverable codes to finite chain rings, generalizing field-based results by developing a TB-type polynomial-interpolation framework over rings. It establishes a ring-aware LRC distance bound and constructs optimal codes using good polynomials, including over Galois rings, with explicit encoding and decoding schemes. The authors further generalize the construction to broader ring settings, remove length limitations via well- and non-well-conditioned sets, and connect maximum possible code lengths to residue-field bounds, offering a path toward longer LRCs in ring alphabets. The results demonstrate the practicality and versatility of ring-based TB constructions for robust distributed storage, while outlining directions for longer codes and richer polynomial families.

Abstract

Locally recoverable codes deal with the task of reconstructing a lost symbol by relying on a portion of the remaining coordinates smaller than an information set. We consider the case of codes over finite chain rings, generalizing known results and bounds for codes over fields. In particular, we propose a new family of locally recoverable codes by extending a construction proposed in 2014 by Tamo and Barg, and we discuss its optimality.
Paper Structure (19 sections, 32 theorems, 69 equations, 1 algorithm)

This paper contains 19 sections, 32 theorems, 69 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Codes over rings and Locality
  3. Generalities on codes over rings
  4. Locally recoverable codes
  5. Lower bound on the minimum distance of a locally recoverable code
  6. Non-existence of $R$-linear codes achieving the LRC bound for certain parameters
  7. Extending the Tamo-Barg construction over finite chain rings
  8. Polynomials over rings
  9. Code construction
  10. Construction of good polynomials over Galois ring
  11. A generalized version of the previous construction
  12. Algebra of good polynomials over finite chain rings
  13. A family of Locally Recoverable Codes
  14. Removing the constraints on code length
  15. Codes over well-conditioned sets with arbitrary length
  16. ...and 4 more sections

Key Result

Theorem 2.4

(Singleton bound) Let $C$ be a code of length $n$ over an alphabet of size $q$. Then

Theorems & Definitions (72)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • proof
  • Definition 2.7
  • Theorem 3.1
  • proof
  • ...and 62 more