Asymptotical properties of linear codes
type: Item
instance of: scholarly article
title: Asymptotical properties of linear codes
zbMATH Open document ID: 1033.94010
DOI: 10.1016/S0012-365X(99)00164-8
published in: Discrete Mathematics
publication date: 2000-03-21T00:00:00Z
review text: 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.
reviewed by: L. M. G. M. Tolhuizen
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 1420371
author: Vladimir Blinovsky
MaRDI profile type: Publication
full work available at URL: s0012-365x(99)00164-8
OpenAlex ID: W2052281551
description: scientific article; zbMATH DE number 1420371