Abstract
The Kneser-Hecke-operator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of self-dual codes of fixed length. It maps a linear self-dual codeC over a finite field to the formal sum of the equivalence classes of those self-dual codes that intersectC in a codi-mension 1 subspace. The eigenspaces of this self-adjoint linear operator may be described in terms of a coding-theory analogue of the Siegel Φ-operator.
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References
C. Bachoc, On harmonic weight enumerators of binary codes.Designs, Codes, and Cryptography 18 (1999), 11–28.
—, Harmonic weight enumerators of non-binary codes and Mac Williams identities. In:Codes and association schemes (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci.56, Amer. Math. Soc, Providence, RI, 2001, pp. 1–23
M. Broué, Codes correcteurs d’erreurs auto-orthogonaux sur le corps à deux éléments et formes quadratiques entières définies positives à discriminant +1.Discrete Math. 17 (1977), no. 3, 247–269.
J. Cannon et al.,The Magma Computational Algebra System for Algebra, Number Theory and Geometry, http://magma .maths .usyd.edu.au/magma/. 90
J. H. Conway andN. J. A. Sloane,Sphere Packings, Lattices and Groups. Springer-Verlag, New York, 1998, 3ed., 1998.
M. Hentschel andA. Krieg, A Hermitian analog of the Schottky form. In:Automorphic forms and zeta functions - Proceedings of the conference in memory of Tsuneo Arakawa - World Scientific, 2006.
M. K. Neser, Klassenzahlen definiter quadratischer Formen.Archiv der Math. 8 (1957), 241–250.
W. Kohnen andR. Salvati Manni, Linear relations between theta series.Osaka J. Math. 41 (2004), 353–356.
G. Nebe, Finite Weil-representations and associated Hecke-algebras. submitted (2006).
G. Nebe, H.-G. Quebbemann, E. M. Rains, andN. J. A. Sloane, Complete weight enumerators of generalized doubly-even self-dual codes.Finite Fields Applic. 10 (2004), 540–550.
—, The invariants of the Clifford groups.Designs, Codes, and Cryptography 24 (2001), 99–121.
—, Codes and invariant theory.Math. Nachrichten 274–275 (2004), 104–116.
---,Self-dual codes and invariant theory. Springer-Verlag, 2006.
G. Nebe andM. Teider, Hecke actions on certain strongly modular genera of lattices.Archiv der Math. 84 (1) (2005), 46–56.
G. Nebe andB. B. Venkov, On Siegel modular forms of weight 12.J. Reine Angew. Math. 531 (2001), 49–60.
M. Oura, The dimension formula for the ring of code polynomials in genus 4.Osaka J. Math. 34(1997), 53–72.
H.-G. Quebbemann, On even codes.Discrete Math. 98 (1991), 29–34.
E. M. Rains andN. J. A. Sloane, Self-dual codes, in: V. S. Pless and W. C. Huffman (eds.),Handbook of Coding Theory. Elsevier, Amsterdam, 1998, pp. 177–294.
B. Runge, Codes and Siegel modular forms.Discrete Math. 148 (1996), 175–204.
N. J. A. Sloane,The On-Line Encyclopedia of Integer Sequences, published electronically at www. research.att.com/∼nj as/sequences/, 2005.
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Communicated by: C. Schweigert