Codes and Siegel modular forms
type: Item
instance of: scholarly article
title: Codes and Siegel modular forms
zbMATH Open document ID: 0854.11071
DOI: 10.1016/0012-365X(94)00271-J
author: Bernhard Runge
published in: Discrete Mathematics
publication date: 1996-03-24T00:00:00Z
review text: This article is mainly concerned with the study of the higher weight enumerators of binary linear codes. If \(C\) is such a code, then its \(g\)-th weight enumerator \(P_g (C)\) is a polynomial in the variables \(X_{\mathbf a}\) for \({\mathbf a}\in \mathbb{F}^g_2\) where the coefficient of a monomial \(X^{\nu_1}_{{\mathbf a}_1} \dots X^{\nu_r}_{{\mathbf a}_r}\) counts the number of \(g\)-tuples \((\alpha_1, \dots, \alpha_g)\in C^g\) such that \({\mathbf a}_i\) gives the coordinates of \((\alpha_1, \dots, \alpha_g)\) in precisely \(\nu_i\) places. A generalized MacWilliams identity is proved for these weight enumerators, and it is shown that the ring generated by the polynomials associated to doubly even self dual codes is the ring of invariants of a certain group \(G_g \subseteq \text{GL} (2^g, \mathbb{C})\) that arises naturally in this context. The trick here is to reduce to the situation of large \(g\) and use the surjectivity of an analogue of Siegel's \(\Phi\)-operator from the theory of modular forms that allows the transition from \(g\) to \(g-1\). Some remarks about theta series associated to lattices constructed from codes are added; for a more detailed treatment of this topic see the work of \textit{R. Salvati-Manni} [Math. Z. 216, 529-539 (1994; Zbl 0814.11029)]. The higher weight enumerators, the associated MacWilliams identities and the connection to theta series have also been studied in the unpublished Diplomarbeit of Herrmann in Bonn (1991) for arbitrary \(\mathbb{F}_q\) instead of \(\mathbb{F}_2\) and by \textit{W. Duke} [Int. Math. Res. Not. 1993, 125-136 (1993; Zbl 0785.94008)].
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 858084
zbMATH Keywords: Siegel modular forms
zbMATH Keywords: higher weight enumerators
zbMATH Keywords: binary linear codes
zbMATH Keywords: generalized MacWilliams identity
zbMATH Keywords: theta series
zbMATH Keywords: lattices
reviewed by: Rainer Schulze-Pillot
MaRDI profile type: Publication
cites work: Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen
cites work: Polynômes des poids de certains codes et fonctions thêta de certains réseaux
cites work: On the enumeration of self-dual codes
cites work: Q4039784
cites work: Q5634146
cites work: Q4771128
cites work: Stabile Modulformen
cites work: Q3963086
cites work: Singular modular forms and theta relations
cites work: Q4773715
cites work: Automorphisms of extra special groups and nonvanishing degree 2 cohomology
cites work: Q3313974
cites work: The biweight enumerator of self-orthogonal binary codes
cites work: Modular Forms and Projective Invariants
cites work: Q5664012
cites work: Q3939880
cites work: Q3966276
cites work: Q4729864
cites work: Q4114031
cites work: Lineare Relationen zwischen Darstellungsanzahlen quadratischer Formen
cites work: Q3820688
cites work: On self-dual, doubly even codes of length 32
cites work: Good self dual codes exist
cites work: Lectures on Curves on an Algebraic Surface. (AM-59)
cites work: Q4043127
cites work: A classification of self-orthogonal codes over GF(2)
cites work: On the classification and enumeration of self-dual codes
cites work: RELATIONS BETWEEN TWO KINDS OF THETA CONSTANTS ON A RIEMANN SURFACE
cites work: Quasihomogeneous singularities
cites work: On Siegel modular forms. Part I.
cites work: On Siegel modular forms part II
cites work: Q4843064
cites work: Hilbert functions of graded algebras
cites work: Invariants of finite groups and their applications to combinatorics
Recommended article: Duality of codes and theta functions.
Recommended article: Q4527064
Recommended article: Complete joint weight enumerators and self-dual codes
Recommended article: Q3820688
Recommended article: Q4272510
Recommended article: Codes over rings, complex lattices and Hermitian modular forms
Recommended article: On harmonic weight enumerators of binary codes
Recommended article: On the notion of Jacobi polynomials for codes
Recommended article: Codes over \(\mathbb{Z}_{2m}\) and Jacobi forms of genus \(n\)
Recommended article: Construction of Jacobi forms from certain combinatorial polynomials
description: scientific article; zbMATH DE number 858084