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Codes and Siegel modular forms. (English) Zbl 0854.11071

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 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enumerators, the associated MacWilliams identities and the connection to theta series have also been studied in the unpublished Diplomarbeit of Herrmann in Bonn (1991) for arbitrary \(\mathbb{F}_q\) instead of \(\mathbb{F}_2\) and by W. Duke [Int. Math. Res. Not. 1993, 125-136 (1993; Zbl 0785.94008)].

MSC:

11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94B05 Linear codes (general theory)
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F27 Theta series; Weil representation; theta correspondences
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