Lattices with theta functions for G(√2) and linear codes

AbstractModular hermitian lattices over Z[i]and, in particular, unimodular lattices over Z[eπi4] give rise to modular forms for Hecke's group G(2)=<(1201), (01−10)>.Two general constructions of such lattices are performed, using codes over F2 and F9. Lattices with an extremal theta-function (i.e., with the largest minimum that Hecke's theory allows) are obtained in C2n for all n < 12, including the densest known sphere-packings of R4n for n = 1, 4, and 8

A representation theorem for algebras with involution

AbstractAlgebras with involution are represented as commutants of two adjoint vector- space endomorphisms

Complete Weight Enumerators of Generalized Doubly-Even Self-Dual Codes

For any q which is a power of 2 we describe a finite subgroup of the group of invertible complex q by q matrices under which the complete weight enumerators of generalized doubly-even self-dual codes over the field with q elements are invariant. An explicit description of the invariant ring and some applications to extremality of such codes are obtained in the case q=4

On even codes

AbstractThe notion of an even selforthogonal code is introduced over Fq, q = 2m, in such a way that codes with this property become ordinary even binary codes (i.e., all weights are multiples of 4) whenFq is identified with Fm2 using aselfcomplementary basis. Extended Reed-Solomon codes of rate ⩽12 turn out to be even. Furthermore, it is shown that asymptotically good even selfdual codes arise from the class field tower method used by Serre to obtain curves with many Fq-rational points