In this paper, we study codes over polynomial rings and give a connection to Jacobi Hilbert modular forms, in particular, Hilbert modular forms over the totally real field via the complete weight enumerators of codes over polynomial rings. / X / 1 / 1 / 1 / scie / scopus
We study codes over a finite field F-4. We relate self-dual codes over F-4 to real 5-mod ular lattices,nd to self-dual codes over F-2 via a Gray map. We construct Jacobi forms over Q(root 5) from the (omplete weight enumerators of self-dual codes over F-4. Furthermore, we relate Hilbert-Siegel forms to the joint weight enumerators of self-dual code...
We introduce the finite ring S-2m = Z(2m) + iZ(2m). We develop a theory of self-dual codes over this ring and relate self-dual codes over this ring to complex unimodular lattices. We describe a theory of shadows for these codes and lattices. We construct a gray map from this ring to the ring Z(2m) and relate codes over these rings, giving special a...
We introduce codes over the ring Z(2m) + alphaZ(2m) + betaZ(2m) + gammaZ(2m). We relate self-dual codes over this ring to quaternionic unimodular lattices and to self-dual codes over Z(2m) via a gray map. We study a connection between the complete weight enumerators of codes over the quaternionic ring Sigma(2m) and Jacobi forms over the half-space ...
We derive formulae for the theta series of the two translates of the even sublattice L-0 of an odd unimodular lattice L that constitute the shadow of L. The proof rests on special evaluations of the Jacobi theta series attached to L and to a certain vector. We produce an analogous theorem for codes. Additionally, we construct non-linear formally se...
We define the complete joint weight enumerator in genus g for codes over Z(2k) and use it to study self-dual codes and their shadows. These weight enumerators are related to the theta series of the associated lattices and Siegel and Jacobi forms are formed from these series. / X / 1 / 1 / 3 / scie / scopus
In this paper we establish a connection between Jacobi forms over a totally real field k=Q(zeta+zeta(-1)), zeta=e(2pii/p), and codes over the field F-p. In particular, we derive a theta series, which is a Jacobi form, from the complete weight enumerator or the Lee weight enumerator of a self-dual code over F-p. / X / 1 / 1 / 3 / scie / scopus
The relations between the complete weight enumerators in genus n of Type II codes over Z(2m) and Jacobi forms of genus n have been discussed. One derives a map between the invariant spaces of the groups G(2m, n) (or H-2m, n, respectively) and the rings of Jacobi forms (or Siegel modular forms, repectively) of genus n. The MacWilliams identities of ...
There has been active research on t-designs constructed from codewords in codes over Z(4). We present an overview of the recent developments in constructing and proving such designs. (C) 2001 Elsevier Science B.V. All rights reserved. / X / 1 / 1 / 4 / scie / scopus
