Extensions of linear codes
R. Hill, P. Lizak · 2002
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.