Constructions of Formally Self-Dual Codes Over $ {\mathbb Z}_{4}$ and Their Weight Enumerators

Abstract

We present three explicit methods for construction of formally self-dual codes over $ {\mathbb Z}_{4}$ . We characterize relations between Lee weight enumerators of formally self-dual codes of length $n$ over $ {\mathbb Z}_{4}$ and those of length $n+2$ ; the first two construction methods are based on these relations. The last construction produces free formally self-dual codes over $ {\mathbb Z}_{4}$ . Using these three constructions, we can find free formally self-dual codes over $ {\mathbb Z}_{4}$ , as well as non-free formally self-dual codes over $ {\mathbb Z}_{4}$ of all even lengths. We find free or non-free formally self-dual codes over $ {\mathbb Z}_{4}$ of lengths up to ten using our constructions. In fact, we obtain 46 inequivalent formally self-dual codes whose minimum Lee weights are larger than self-dual codes of the same length. Furthermore, we find 19 non-linear extremal binary formally self-dual codes of lengths 12, 16, and 20, up to equivalence, from formally self-dual codes over $ {\mathbb Z}_{4}$ by using the Gray map.