Some basic theory on the duality of codes over two non-unital rings of order $ 6 $, namely $ H_{23} $ and $ H_{32} $ is presented. For a code $ {{\mathcal C}} $ over these rings, there is an associated binary code $ {{\mathcal C}}_a $ and a ternary code $ {{\mathcal C}}_b $. Self-orthogonal, self-dual, and quasi self-dual (QSD) codes over these rings are characterized using the associated codes $ {{\mathcal C}}_a $ and $ {{\mathcal C}}_b $, and a classification of self-orthogonal codes for short lengths is given. In addition, a building-up construction for self-orthogonal codes is presented, and cyclic and linear complementary dual (LCD) codes over the said rings are introduced.
Citation: Altaf Alshuhail, Rowena Alma Betty, Lucky Galvez. Duality of codes over non-unital rings of order six[J]. AIMS Mathematics, 2025, 10(8): 18784-18800. doi: 10.3934/math.2025839
Some basic theory on the duality of codes over two non-unital rings of order $ 6 $, namely $ H_{23} $ and $ H_{32} $ is presented. For a code $ {{\mathcal C}} $ over these rings, there is an associated binary code $ {{\mathcal C}}_a $ and a ternary code $ {{\mathcal C}}_b $. Self-orthogonal, self-dual, and quasi self-dual (QSD) codes over these rings are characterized using the associated codes $ {{\mathcal C}}_a $ and $ {{\mathcal C}}_b $, and a classification of self-orthogonal codes for short lengths is given. In addition, a building-up construction for self-orthogonal codes is presented, and cyclic and linear complementary dual (LCD) codes over the said rings are introduced.
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Altaf Alshuhail, Rowena Alma Betty, Lucky Galvez. Duality of codes over non-unital rings of order six[J]. AIMS Mathematics, 2025, 10(8): 18784-18800. doi: 10.3934/math.2025839
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