Scholarlink

<p>Algebraic geometry codes (AG codes), also known as geometric Goppa codes over finite fields, have a unique characteristic: their parameters can be bounded using the degree of certain divisors, which allows for a clear description of the codes. Although the theoretical foundation of this subject is quite complex, AG codes demonstrate asymptotically favorable parameters. They were the first linear codes to exceed the Gilbert-Varshamov bound. Additionally, it is well established that AG codes of genus zero are Maximum Distance Separable (MDS) codes. In this paper, we explore the concept of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation>-dimensional linear intersection pair (referred to as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation>-DLIP) of codes, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation> is a positive integer. The <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation>-dimensional linear intersection pair is derived from the intersection of two linear codes whose intersection has dimension. The <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation>-dimension of the intersection of two linear codes represents the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation>-dimensional linear intersection pair. Specifically, we investigate the properties of these pairs within the context of AG codes. We provide several characterizations of pairs of AG codes that can form a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation>-DLIP. Furthermore, we explore various constructions of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\ell\)</EquationSource> </InlineEquation>-DLIPs for algebraic geometry codes, primarily deriving them from Kummer extensions, hyperelliptic function fields, and elliptic curves.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On \(\ell\)-dimensional linear intersection pairs of algebraic geometry codes

  • 期刊论文

  • 发表日期：

    2025年10月7日

查看全文
  • Sanjit Bhowmick,
  • Kuntal Deka,
  • Sihem Mesnager

摘要

Algebraic geometry codes (AG codes), also known as geometric Goppa codes over finite fields, have a unique characteristic: their parameters can be bounded using the degree of certain divisors, which allows for a clear description of the codes. Although the theoretical foundation of this subject is quite complex, AG codes demonstrate asymptotically favorable parameters. They were the first linear codes to exceed the Gilbert-Varshamov bound. Additionally, it is well established that AG codes of genus zero are Maximum Distance Separable (MDS) codes. In this paper, we explore the concept of the \(\ell\) -dimensional linear intersection pair (referred to as \(\ell\) -DLIP) of codes, where \(\ell\) is a positive integer. The \(\ell\) -dimensional linear intersection pair is derived from the intersection of two linear codes whose intersection has dimension. The \(\ell\) -dimension of the intersection of two linear codes represents the \(\ell\) -dimensional linear intersection pair. Specifically, we investigate the properties of these pairs within the context of AG codes. We provide several characterizations of pairs of AG codes that can form a \(\ell\) -DLIP. Furthermore, we explore various constructions of \(\ell\) -DLIPs for algebraic geometry codes, primarily deriving them from Kummer extensions, hyperelliptic function fields, and elliptic curves.