Small additive quaternary codes.
Consider the additive group \(Q= \mathbb{Z}_2\times \mathbb{Z}_2\). A code with codewords of length \(n\) over \(Q\) is called additive if it is an additive subgroup of \(Q^n\). A code with codewords of length \(n\) over \(Q\) is called optimal when it has the largest possible size for given values of \(n\) and \(d\). In this article, the authors determine the parameters of the optimal additive codes of word length at most 12 over \(Q\), up to two specific sets of parameters. Some of the codes found improve on the best quaternary codes known. This search for optimal additive codes over \(Q\) is equivalent to the search for how many lines it is possible to take in a binary projective space such that any \(t\) are independent, or equivalently, how many lines it is possible to take in a binary projective space such that no hyperplane contains more than \(m\) of them. Here, lines may be repeated, and a line is sometimes the span of two not necessarily distinct points.
scientific article; zbMATH DE number 2056151