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<p>Paley-type partial difference sets and skew–Hadamard difference sets are classical objects in algebraic combinatorics, known for their rich connections with graph theory, coding theory, and group theory. In this paper, we explore new links between these combinatorial structures and group codes arising as ideals in finite group algebras. We construct such codes from difference sets and determine their dimensions in several cases. As an application of our links, we explicitly compute the full set of primitive central idempotents in certain abelian <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( p \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-group algebras, by employing the classical sets of quadratic residues and non-residues modulo <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( p \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>, which are well-studied examples of difference and partial difference sets—we also obtain their dimensions and estimate their minimum weights.</p>

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On group codes arising from Paley-type partial difference sets and skew–Hadamard difference sets

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月12日

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  • Vitor Araujo Garcia

摘要

Paley-type partial difference sets and skew–Hadamard difference sets are classical objects in algebraic combinatorics, known for their rich connections with graph theory, coding theory, and group theory. In this paper, we explore new links between these combinatorial structures and group codes arising as ideals in finite group algebras. We construct such codes from difference sets and determine their dimensions in several cases. As an application of our links, we explicitly compute the full set of primitive central idempotents in certain abelian \( p \) p -group algebras, by employing the classical sets of quadratic residues and non-residues modulo \( p \) p , which are well-studied examples of difference and partial difference sets—we also obtain their dimensions and estimate their minimum weights.