An Extension Theorem for Linear Codes

One of the first results one meets in coding theory is that a binary linear n,k,d] code, whose minimum distance 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 generalise to q-ary codes. 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. Applications will be given to the construction and classification of good codes, to proving the non- existence of certain codes, and also an application in finite geometry.