Asymptotically exact uniform bounds for spectra of cosets of linear codes (Q917523)

Let C be a binary linear (n,k)-code and let \(L_ C(y,r)\) be the number of codewords of weight r in the coset \(y+C\). Then the following theorem is proved: Theorem. For all \(n\in {\mathbb{N}}\) and all binary vectors \(y\in {\mathbb{F}}^ n_ 2\) at least a fraction of \(1-2^{-n\cdot o(n)}\) of all binary linear (n,k)-codes satisfies the inequality \(k/n\leq 1- H(r/n)+\log_ 2L_ C(y,r)+o(1).\) (Here \(H(\alpha):=-\alpha \cdot \log_ 2\alpha -(1-\alpha)\cdot \log_ 2(1-\alpha)\) denotes the binary entropy function.) From this theorem the author derives an asymptotic Goblick-bound on the covering radius for the same fraction of \(1-2^{- n\cdot o(n)}\) of all binary linear (n,k)-codes.