Expressing the minimum distance, weight distribution and covering radius of codes by means of the algebraic and numerical normal forms of their indicators

Abstract

<p style='text-indent:20px;'>We consider the algebraic normal form (ANF) of the indicators (i.e. characteristic functions) of linear binary codes, and characterize the minimum distance of such codes in a very simple way by means of this ANF. We extend this characterization to nonlinear binary codes, via another representation, the numerical normal form (NNF). We further extend these characterizations to linear codes over finite fields and (after introducing a generalization of the NNF to functions from <inline-formula><tex-math id="M1">\begin{document}$ {\Bbb F}_p^n $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M2">\begin{document}$ {\Bbb R} $\end{document}</tex-math></inline-formula>) to unrestricted codes over prime fields. We also study the weight distribution by means of the NNF, and the covering radius of binary codes with the same approach; the latter is more difficult to address, but we obtain some results as well.</p>