On Additive GF(4) Codes
Philippe Gaborit, W. Cary Huffman, Jon-Lark Kim, Vera S. Pless · 2001
. In this paper we discuss the status of the classication of the extremal additive codes over GF (4) which are self-dual under the trace inner product. These codes are of interest because of their connection to quantum codes. There are 5 inequivalent codes of length 8, 8 of length 9, at least 56 of length 10, and exactly 1 of length 11. We discuss the methods of classication, the concepts of lengthening and shortening, and connections to binary codes. 1. Introduction An additive code C over GF (4) of length n is an additive subgroup of GF (4) n . As C is a free GF(2)-module, it has size 2 k for some 0 k 2n. We call C an (n; 2 k ) code. It has a basis, as a GF(2)-module, consisting of k basis vectors; a generator matrix of C will be a k n matrix with entries in GF (4) whose rows are a basis of C. Interest in additive codes over GF (4) has arisen because of their correspondence to quantum codes as described in [3]. There is a natural inner product arising from the trace m...