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<p>In this article, we consider the decoding problem of affine Grassmann codes over nonbinary fields. We use matrices of different ranks to construct a large set consisting of parity checks of affine Grassmann codes, which are orthogonal with respect to some fixed coordinate. By leveraging the automorphism groups of these codes, we generate a set of orthogonal parity checks for each coordinate. Using these parity checks, we perform majority logic decoding to correct a large number of errors in affine Grassmann codes. The order of error correction capability and the complexity of this decoder for affine Grassmann codes are the same as those of the majority logic decoder for Grassmann codes proposed in [<CitationRef CitationID="CR4">4</CitationRef>]. Ultimately, we provide an example of this decoder for the ternary affine Grassmann code <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({C^{\mathbb A}(2,4)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mi mathvariant="double-struck">A</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Majority logic decoding of affine Grassmann codes over nonbinary fields

  • 期刊论文

  • 发表日期：

    2026年7月28日

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  • Fernando Piñero González,
  • Prasant Singh,
  • Rohit Yadav

摘要

In this article, we consider the decoding problem of affine Grassmann codes over nonbinary fields. We use matrices of different ranks to construct a large set consisting of parity checks of affine Grassmann codes, which are orthogonal with respect to some fixed coordinate. By leveraging the automorphism groups of these codes, we generate a set of orthogonal parity checks for each coordinate. Using these parity checks, we perform majority logic decoding to correct a large number of errors in affine Grassmann codes. The order of error correction capability and the complexity of this decoder for affine Grassmann codes are the same as those of the majority logic decoder for Grassmann codes proposed in [4]. Ultimately, we provide an example of this decoder for the ternary affine Grassmann code \({C^{\mathbb A}(2,4)}\) C A ( 2 , 4 ) .