Manabu Oura

Type II codes, even unimodular lattices, and invariant rings

We study self-dual codes over the ring Z/sub 2k/ of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over Z/sub 2k/ which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extremal Type II code over Z/sub 6/ of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over Z/sub 2k/ are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over Z/sub 2k/ and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over Z/sub 2k/ are also introduced.

Higher weights and graded rings for binary self-dual codes

The theory of higher weights is applied to binary self-dual codes. Bounds are given for the second minimum higher weight and a Gleason-type theorem is derived for the second higher weight enumerator. The second weight enumerator is shown to be unique for the putative [72, 36, 16] Type II code and the first three minimum weights are computed for optimal codes of length less than 32. We also determine the structures of the graded rings associated with the code polynomials of higher weights for small genera, one of which has the property that it is not Cohen-Macaulay.

The dimension formula for the ring of code polynomials in genus 4

The purpose of this paper is to study the dimension formula of the invariant ring of the specified group H5. This ring appears in the theory of Siegel modular forms and in coding theory. As an application of our dimension formula we give another proof of the fact the associated g-th weight enumerators of the 9 self-dual doubly-even codes of length 24 are linearly independent if and only if g ≥ 6, which is proved in a recent paper by Oura-Poor-Yuen.

Note on the g-fold Joint Weight Enumerators of Self-Dual Codes over ℤ k

Abstract. Recently there has been interest in self-dual codes over finite rings. In this note, g-fold joint weight enumerators of codes over the ring ℤk of integers modulo k are introduced as a generalization of the biweight enumerators. We establish the MacWilliams relations for these weight enumerators and investigate the biweight enumerators of self-dual codes over ℤk. We derive Gleason-type theorems for the corresponding biweight enumerators with the help of invariant theory.

TOWARDS THE SIEGEL RING IN GENUS FOUR

Runge gave the ring of genus three Siegel modular forms as a quotient ring, R3/〈J(3)〉. R3 is the genus three ring of code polynomials and J(3) is the difference of the weight enumerators for the e8 ⊕ e8 and d+16 codes. Freitag and Oura gave a degree 24 relation, R (4) 0 , of the corresponding ideal in genus four; R (4) 0 is also a linear combination of weight enumerators. We take another step toward the ring of Siegel modular forms in genus four. We explain new techniques for computing with Siegel modular forms and actually compute six new relations, classifying all relations through degree 32. We show that the local codimension of any irreducible component defined by these known relations is at least 3 and that the true ideal of relations in genus four is not a complete intersection. Also, we explain how to generate an infinite set of relations by symmetrizing first order theta identities and give one example in degree 32. We give the generating function of R5 and use it to reprove results of Nebe [25] and Salvati Manni [37]. §

A theta relation in genus 4

The 2 g theta constants of second kind of genus g generate a graded ring of dimension g(g + 1)/2. In the case g ≥ 3 there must exist algebraic relations. In genus g = 3 it is known that there is one defining relation. In this paper we give a relation in the case g = 4. It is of degree 24 and has the remarkable property that it is invariant under the full Siegel modular group and whose Φ-image is not zero. Our relation is obtained as a linear combination of code polynomials of the 9 self-dual doubly-even codes of length 24.

Higher Weights for Ternary and Quaternary Self-Dual Codes*

We study higher weights applied to ternary and quaternary self-dual codes. We give lower bounds on the second higher weight and compute the second higher weights for optimal codes of length less than 24. We relate the joint weight enumerator with the higher weight enumerator and use this relationship to produce Gleason theorems. Graded rings of the higher weight enumerators are also determined.

The joint weight enumerators and Siegel modular forms

The weight enumerator of a binary doubly even self-dual code is an isobaric polynomial in the two generators of the ring of invariants of a certain group of order 192. The aim of this note is to study the ring of coefficients of that polynomial, both for standard and joint weight enumerators.

On the cycle index and the weight enumerator

In this paper, we introduce the concept of the complete cycle index and discuss a relation with the complete weight enumerator in coding theory. This work was motivated by Cameron’s lecture notes “Polynomial aspects of codes, matroids and permutation groups.”

Centralizer algebras of the group associated to Z4-codes

Abstract The purpose of this paper is to investigate the finite group which appears in the study of the Type II Z 4 -codes. To be precise, it is characterized in terms of generators and relations, and we determine the structure of the centralizer algebras of the tensor representations of this group.