Type \(I\) Codes over \(GF(4)\)

Abstract

It was shown by Gaborit et al. [10] that a Euclidean self-dual code over \({GF}(4)\) with the property that there is a codeword whose Lee weight \(\equiv 2 \pmod{4}\) is of interest because of its connection to a binary singly-even self-dual code. Such a self-dual code over \({GF}_4\) is called Type I. The purpose of this paper is to classify all Type I codes of lengths up to 10 and extremal Type I codes of length 12, and to construct many new extremal Type I codes over \({GF}(4)\) of lengths from 14 to 22 and 34. As a byproduct, we construct a new extremal singly-even self-dual binary [36, 18, 8] code, and a new extremal singly-even self-dual binary [68, 34, 12] code with a previously unknown weight enumerator \(W_2\) for \(\beta = 95\) and \(\gamma = 1\).