Extensions of linear codes
Extensions of linear codes
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线性码的扩展
DOI：
10.1109/isit.1995.550332
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
P. Lizak
中科院分区：
文献类型：
--
作者：
R. Hill;P. Lizak
在编码理论中遇到的第一个结果是，最小权值为奇数的二元线性[n，k，d]码可以扩展为[n+1，k，d+1]码。这是关于二进制码的少数几个不能明显推广到q元码的基本结果之一。虽然我们可以很容易地通过增加校验位来扩展q位码，但是在什么情况下这样的扩展会增加最小距离是不清楚的。本文的目的是给出q元[n，k，d]码可扩展为[n+1，k，d+1]码的一个简单充分条件。该结果实际上是上述结果对二进制码的推广。它也推广了由van Eupen和Lisonek提出的三进制码的结果，他们的证明使用了二次形式。这个推广有一个初等证明。
One of the first results one meets in coding theory is that a binary linear [n,k,d]-code, whose minimum weight is odd, can be extended to an [n+1,k,d+1]-code. This is one of the few elementary results about binary codes which does not obviously generalize to q-ary codes. Although one can readily extend a q-ary code, by adding a further check digit, it is not clear under what circumstances such an extension will increase the minimum distance. The aim of this paper is to give a simple sufficient condition for a q-ary [n,k,d]-code to be extendable to an [n+1,k,d+1]-code. The result is indeed a generalization of the above result for binary codes. It also generalizes a result for ternary codes due to van Eupen and Lisonek, whose proof made use of quadratic form. The present generalization has an elementary proof.