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<p>In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes (GAG codes) using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter <i>r</i> in such codes by using only non-rational places of degree at most <i>r</i>. 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1597_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb F_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>. We then investigate similarities with some 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 with GAG codes 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 approach with GAG codes.</p>

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Introducing locality in some generalized AG codes

  • 期刊论文

  • 发表日期：

    2025年2月24日

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  • Bastien Pacifico

摘要

In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes (GAG 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 degree 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 \(\mathbb F_3\) F 3 . We then investigate similarities with some 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 with GAG codes 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 approach with GAG codes.