Asymptotical properties of linear codes
The author collects some of his previous results on some asymptotic properties of linear codes and formulates some problems. (1) Give estimates for the fraction of \([n,k]\)-codes for which the covering radius \(e\) satisfies \({k\over n}>1-h(e/n)+O(1)\) for various ``functions'' satisfying \(O(1) \), \(n\to\infty\) \((h\) is the binary entropy function). (2) By a random coding argument, the author has shown that there is a \(\kappa >0\) such that for all \(k\) and \(n\) there exists an \([n,k]\)-code \(C\) such that the number of words of weight \(i\), \(A_i(C)\), satisfies \(A_i(C) \leq(K \sqrt{n\ln n}+ 1)2^{k-n} {n\choose i}\) for \(0\leq i\leq n\). He now wonders if the term \(K\sqrt{n\ln n}\) can be improved to \(K\sqrt n\). A similar question arises in the closely related problem of bounding the maximum undetected error probability. (3) A linear code is called list-of-\(L\) \(t\) error correcting if the spheres with radius \(t\) centered in more than \(L\) distinct codewords have an empty intersection. An open problem is to find an asymptotic existence bound for linear list-of-\(L\) codes of length \(n\) correcting \(\tau\cdot n\) errors, where \(L\) and \(\tau\) are fixed and \(n\) tends to infinity.
scientific article; zbMATH DE number 1420371