EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

Abstract

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ 4 -codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ 4 -codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽 4 -codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.