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<p>In this paper, we study two-dimensional (2-D) skew cyclic codes over the Galois ring <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(R=GR(4^m)\)</EquationSource> </InlineEquation>. First, we briefly discuss skew cyclic codes over <i>R</i>, and then we define a consistent set for left ideals in a bi-variate skew polynomial ring over <i>R</i> to study 2-D skew cyclic codes. We provide a necessary and sufficient condition for a generating set to be a consistent set. Thereafter, we define an algorithm to construct a consistent set starting from an arbitrary generating set. Finally, we demonstrate the use of these consistent sets to study 2-D skew cyclic codes. Additionally, we present a set of two 2-D skew cyclic codes over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(GF(2^m)\)</EquationSource> </InlineEquation> associated with a 2-D skew cyclic code over <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(GR(4^m)\)</EquationSource> </InlineEquation>.</p>

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2-D skew cyclic codes over the Galois ring \(GR(4^m)\)

  • 期刊论文

  • 发表日期：

    2026年3月28日

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  • Saumya Shah,
  • Amit Sharma

摘要

In this paper, we study two-dimensional (2-D) skew cyclic codes over the Galois ring \(R=GR(4^m)\) . First, we briefly discuss skew cyclic codes over R, and then we define a consistent set for left ideals in a bi-variate skew polynomial ring over R to study 2-D skew cyclic codes. We provide a necessary and sufficient condition for a generating set to be a consistent set. Thereafter, we define an algorithm to construct a consistent set starting from an arbitrary generating set. Finally, we demonstrate the use of these consistent sets to study 2-D skew cyclic codes. Additionally, we present a set of two 2-D skew cyclic codes over \(GF(2^m)\) associated with a 2-D skew cyclic code over \(GR(4^m)\) .