Codes with given distances (Q1094390)

Let \(F\) be the field with two elements and \(C\) be a code of length \(n\) over \(F\), i.e. a subset of the \(n\)-dimensional vector space over \(F\). Let \(D(C)=\{d(x,y),\ x,y\in C,\ x\ne y\}\), where \(d(x,y)\) denotes the Hamming distance between \(x\) and \(y\). This paper concerns several problems about codes for which \(D(C)\) does not contain some prescribed values. One of the main results says that if \(C\) is linear of length \(4t\) and dimension at least \(2t\), then \(2t-1\) or \(2t\) belongs to \(D(C)\). Using this result the authors give a short proof for Gleason's theorem; if \(C\) is a linear code of length \(n\) which is self-dual and \(4\mid d\) for every \(d\in D(C)\), then \(8\mid n\). Another consequence of the main result is that if \(F_1,\ldots, F_{2t-1}\) are subsets of \(X\), \(| X| =4t\), then there exists a \(G\le X\) of size \(2t\) such that \(| G\cap F_i|\) is even for all \(i\), \(1\le i\le 2t-1\). For the proof of the main results a lemma introducing the binormal forms of binary matrices is needed.