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<p>Given a simple, connected graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal G\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation>, let <i>C</i> denote the binary linear code whose generator matrix is obtained by appending the incidence matrix of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal G\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> to the identity matrix. In this paper, we establish a bijection between minimal codewords in <i>C</i> and the non-equivalent walks in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal G\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation>. For several families of graphs, we determine the exact number of minimal codewords.</p>

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Minimal binary codewords derived from the incidence-matrix approach

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • Boran Kim

摘要

Given a simple, connected graph \(\mathcal G\) G , let C denote the binary linear code whose generator matrix is obtained by appending the incidence matrix of \(\mathcal G\) G to the identity matrix. In this paper, we establish a bijection between minimal codewords in C and the non-equivalent walks in \(\mathcal G\) G . For several families of graphs, we determine the exact number of minimal codewords.