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Bounds on Minimal Codewords in Linear Codes (1994)
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BibTeX
@MISC{Lüders-Jensen94boundson,
author = {Kim Lüders-Jensen and Thomas Jakobsen},
title = {Bounds on Minimal Codewords in Linear Codes},
year = {1994}
}
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Abstract
The notion of minimal codewords in linear codes was introduced recently by Massey. In this paper two weight bounds on minimal codewords are proved; an upper bound above which no codewords are minimal and a lower bound below which all codewords are minimal. It is shown for Hamming codes that every weight class between the two bounds contains at least one minimal codeword and at least one non-minimal codeword. 1 Introduction In the following we will consider a linear (n; k; d) code, C, over GF (q) with generator matrix G. ff is a primitive element in GF (q). Given a codeword, c 2 C, its Hamming weight is denoted w(c). G c will denote the submatrix consisting of the columns in G, where the value of the corresponding component in c is 0, and ae c denotes the rank of G c . We will say that a non-zero word is normalized if the leftmost non-zero component is 1. As in [1] a word, c, is said to cover another word, c 0 , if for each position where c 0 is non-zero, c is non-zero as well. Or ...
