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<p>We define a family of rings of size <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(2^{5^k}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <msup> <mn>5</mn> <mi>k</mi> </msup> </msup> </math></EquationSource> </InlineEquation> and for each ring in the family we give <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2^k\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> linear, orthogonality preserving, Gray maps to the binary space. We prove that the image of a self-dual code is self-dual and the image of an LCD code is LCD. Moreover, we show that the images under these Gray maps have a rich automorphism group. We relate the shadows of binary codes to additive codes over rings in this family. We also construct binary LCD codes as images of codes over the rings of order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(2^5\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mn>5</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(2^{25}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mn>25</mn> </msup> </math></EquationSource> </InlineEquation> in the family.</p>

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Codes over an infinite family of local rings of order \(2^{5^k}\) with two Gray maps

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Steven T. Dougherty,
  • Joe Gildea,
  • Adrian Korban,
  • Adam M. Roberts

摘要

We define a family of rings of size \(2^{5^k}\) 2 5 k and for each ring in the family we give \(2^k\) 2 k linear, orthogonality preserving, Gray maps to the binary space. We prove that the image of a self-dual code is self-dual and the image of an LCD code is LCD. Moreover, we show that the images under these Gray maps have a rich automorphism group. We relate the shadows of binary codes to additive codes over rings in this family. We also construct binary LCD codes as images of codes over the rings of order \(2^5\) 2 5 and \(2^{25}\) 2 25 in the family.