<p>We study a non-unital and non-commutative ring <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_{m}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, called ring of ordered sum over a ring <i>R</i>. We obtain linear codes over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S_{m}(\mathbb {F}_{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, also known as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>-codes, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {F}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> is the binary field. The algebraic structure of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$S_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>-codes, particularly, their residue and torsion codes, will be explored. Moreover, a generalized notion of quasi self-dual codes will be introduced. Finally, we give results on the weight enumerators of these codes and construct <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$S_{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-codes in short lengths using the residue and torsion codes.</p>

Quasi self-dual codes over $S_{m}(\mathbb {F}_{2})$

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发表日期：

2026年1月21日

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Jupiter Pilongo,
Rowena Alma Betty,
Lucky Erap Galvez

摘要

We study a non-unital and non-commutative ring $S_{m}(R)$ $S_{m} (R)$ , called ring of ordered sum over a ring R. We obtain linear codes over $S_{m}(\mathbb {F}_{2})$ $S_{m} (F_{2})$ , also known as $S_{m}$ $S_{m}$ -codes, where $\mathbb {F}_{2}$ $F_{2}$ is the binary field. The algebraic structure of $S_{m}$ $S_{m}$ -codes, particularly, their residue and torsion codes, will be explored. Moreover, a generalized notion of quasi self-dual codes will be introduced. Finally, we give results on the weight enumerators of these codes and construct $S_{3}$ $S_{3}$ -codes in short lengths using the residue and torsion codes.

Quasi self-dual codes over \(S_{m}(\mathbb {F}_{2})\)

摘要