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On the notion of Jacobi polynomials for codes. (English) Zbl 0888.94028

The connection between theta functions and weight enumerators is well known [see e.g. W. Ebeling, Lattices and codes. Braunschweig: Vieweg Verlag (1994; Zbl 0805.11048)]. The author introduces Jacobi polynomials in correspondence to Jacobi theta functions [see M. Eichler and D. Zagier, The theory of Jacobi forms. Boston etc.: Birkhäuser Verlag (1985; Zbl 0554.10018)]. The Jacobi polynomial of the code \(C\subseteq \mathbb F_q^n\) with respect to \(v\in \mathbb F_q^n\) is the polynomial \[ \text{Jac} (C,v\mid X,Z):= \sum_{u\in C}X^{u*u} Z^{u*v}, \] where \(u*v\) is number of components of \(u,v\) with \(u_iv_i\neq 0\). The author proves an identity which generalizes the MacWilliams identity and some further properties of these polynomials.

MSC:

94B05 Linear codes (general theory)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11F50 Jacobi forms
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