Self-dual codes over GF(4)

Abstract

This paper studies codes $\text{C}$ over GF(4) which have even weights and have the same weight distribution as the dual code C^⊥. Some of the results are as follows. All such codes satisfy $\text{C}{}^{\mathrm{\perp }}\text{=}\text{C}$ (If C^⊥= C, $\text{T}$ has a binary basis.) The number of such C's is determined, and those of length ⩽14 are completely classified. The weight enumerator of $\text{C}$ is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for $\text{C}$ to be extended cyclic. Two new 5-designs are constructed. A generator matrix for $\text{C}$ can be taken to have the form [I | B], where $\text{B}{}^{\mathrm{\perp }}\text{=}\text{B}$. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.