On spectra of linear codes (Q2190889)

One of the most famous equalities in coding theory is the McWilliams equality giving a relation between the spectrum \(\{a_0,\ldots,a_n\}\) of a vector space \(V\) and \(\{b_0,\ldots,b_n\}\) of its orthogonal subspace \(V^\ast.\) In this work, V. K. Leont'ev presents a MacWilliams-type equality by analyzing the behavior of the sequence \(\left\{a_s/\binom{n}{s}\right\}.\) With \(\varphi_s(n,i)\) being the coefficient of \(z^n\) in the expansion of \((1+z)^{n-i}(1-z)^i=\sum\limits_{s=0}^{n}\varphi_s(n,i)z^s\) (the Krawtchouk polynomial) equalities for \(\sum\limits_{s=0}^{n}\cfrac{\varphi_s(n,i)}{\binom{n}{s}}\) and \(\sum\limits_{s=0}^{n}\cfrac{a_s}{\binom{n}{s}}\) are devised. As a consequence, an asymptotic inequality for the expectation \(E\beta\) for \(\beta=\frac{1}{|G|}\sum\limits_{j=1}^{n}\cfrac{2^na_j^2}{\binom{n}{j}}\) (where \(\{a_0,\dots,a_n\}\) is the spectrum of a code \(G\)) is shown which answers a question posed by Sidel'nikov.