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<p>We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. To the best of our knowledge, this is the first work in which the dual and hull codes are studied in the framework of the two generic constructions; in particular, we propose a Gram–Schmidt process to compute them explicitly. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> to belong in the dual. This achievement was generally obtained when the functions were weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals.</p>

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Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions

  • 期刊论文

  • 发表日期：

    2025年4月15日

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  • Virginio Fratianni

摘要

We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. To the best of our knowledge, this is the first work in which the dual and hull codes are studied in the framework of the two generic constructions; in particular, we propose a Gram–Schmidt process to compute them explicitly. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of \(\mathcal {C}\) C to belong in the dual. This achievement was generally obtained when the functions were weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals.