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<p>A code is locally recoverable when each symbol in one of its codewords can be reconstructed as a function of <i>r</i> other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that is, evaluation codes where each codeword symbol can be reconstructed from several disjoint sets of other symbols. The simplest case, where the code’s underlying variety is a plane, exhibits noteworthy properties: When <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(r = 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, 2, 3, they are optimal; when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(r \ge 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, they are optimal with probability approaching 1 as the alphabet size grows. Additionally, their information rate is close to the theoretical limit. In higher dimensions, our codes form a family of asymptotically good codes.</p>

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Locally recoverable algebro-geometric codes from projective bundles

  • 期刊论文

  • 发表日期：

    2026年2月20日

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  • Konrad Aguilar,
  • Angelynn Álvarez,
  • René Ardila,
  • Pablo S. Ocal,
  • Cristian Rodriguez Avila,
  • Anthony Várilly-Alvarado

摘要

A code is locally recoverable when each symbol in one of its codewords can be reconstructed as a function of r other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that is, evaluation codes where each codeword symbol can be reconstructed from several disjoint sets of other symbols. The simplest case, where the code’s underlying variety is a plane, exhibits noteworthy properties: When \(r = 1\) r = 1 , 2, 3, they are optimal; when \(r \ge 4\) r ≥ 4 , they are optimal with probability approaching 1 as the alphabet size grows. Additionally, their information rate is close to the theoretical limit. In higher dimensions, our codes form a family of asymptotically good codes.