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Introducing locality in some generalized AG codes

Bastien Pacifico

TL;DR

It is shown that one can obtain a locality parameter r by using only non-rational places of degree at most r in GAG codes by using only non-rational places of degree at most r, a new way to construct locally recoverable codes (LRCs).

Abstract

In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter $r$ in such codes by using only non-rational places of degrees at most $r$. This is, up to the author's knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over $\mathbb F_3$. We then investigate similarities with certain concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia-Stichtenoth tower of function fields, for both our construction and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our new approach.

Introducing locality in some generalized AG codes

TL;DR

It is shown that one can obtain a locality parameter r by using only non-rational places of degree at most r in GAG codes by using only non-rational places of degree at most r, a new way to construct locally recoverable codes (LRCs).

Abstract

In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter in such codes by using only non-rational places of degrees at most . This is, up to the author's knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over . We then investigate similarities with certain concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia-Stichtenoth tower of function fields, for both our construction and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our new approach.
Paper Structure (5 sections, 1 theorem, 4 equations)

This paper contains 5 sections, 1 theorem, 4 equations.

Table of Contents

  1. Introduction
  2. Known results
  3. Contributions and organization
  4. Locally Recoverable Codes (LRCs)
  5. Generalities

Key Result

theorem thmcountertheorem

Let $\mathcal{C}$ be a $q-$ary linear code with parameters $[n,k,d]$ with locality $r$. The rate of $\mathcal{C}$ verifies The minimum distance $d$ of $\mathcal{C}$ verifies

Theorems & Definitions (2)

  • definition thmcounterdefinition
  • theorem thmcountertheorem