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<p>We present enumeration formulas for double circulant (DC) codes of length <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2m\)</EquationSource> </InlineEquation> and four circulant (FC) codes of length <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(4m\)</EquationSource> </InlineEquation> over the finite field <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {F}_q\)</EquationSource> </InlineEquation>, with prescribed Euclidean hull dimension, assuming <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\gcd (m, q) = 1\)</EquationSource> </InlineEquation>. These formulas significantly generalize previous results that were limited to special cases. In particular, we resolve an open problem posed by Zhu and Shi (J. Appl. Math. Comput. 68:1227–1244, 2022) concerning the case of reciprocal pair factors in the factorization of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(x^m - 1\)</EquationSource> </InlineEquation> for the enumeration of self-dual and linear complementary dual (LCD) FC codes. Additionally, we correct an enumeration formula used in Zhu and Shi (Bull. Aust. Math. Soc. 98(1):159–166, 2018), leading to an improved bound on the relative distance for LCD FC codes.</p>

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Asymptotic performance of double circulant and four circulant codes with small hull dimension

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月8日

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  • Zohreh Aliabadi,
  • Tekgül Kalaycı,
  • Mohammad Zadehdabbagh

摘要

We present enumeration formulas for double circulant (DC) codes of length \(2m\) and four circulant (FC) codes of length \(4m\) over the finite field \(\mathbb {F}_q\) , with prescribed Euclidean hull dimension, assuming \(\gcd (m, q) = 1\) . These formulas significantly generalize previous results that were limited to special cases. In particular, we resolve an open problem posed by Zhu and Shi (J. Appl. Math. Comput. 68:1227–1244, 2022) concerning the case of reciprocal pair factors in the factorization of \(x^m - 1\) for the enumeration of self-dual and linear complementary dual (LCD) FC codes. Additionally, we correct an enumeration formula used in Zhu and Shi (Bull. Aust. Math. Soc. 98(1):159–166, 2018), leading to an improved bound on the relative distance for LCD FC codes.