144 research outputs found
The invariants of the Clifford groups
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not
3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an
extraspecial group of order 2^(1+2m) extended by an orthogonal group). This
group and its complex analogue CC_m have arisen in recent years in connection
with the construction of orthogonal spreads, Kerdock sets, packings in
Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs.
In this paper we give a simpler proof of Runge's 1996 result that the space
of invariants for C_m of degree 2k is spanned by the complete weight
enumerators of the codes obtained by tensoring binary self-dual codes of length
2k with the field GF(2^m); these are a basis if m >= k-1. We also give new
constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix
[2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power
of M, and C_m is the automorphism group of this tensor power. Also, if C is a
binary self-dual code not generated by vectors of weight 2, then C_m is
precisely the automorphism group of the complete weight enumerator of the
tensor product of C and GF(2^m). There are analogues of all these results for
the complex group CC_m, with ``doubly-even self-dual code'' instead of
``self-dual code''.Comment: Latex, 24 pages. Many small improvement
Kneser-Hecke-operators in coding theory
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 code over a finite field to the
formal sum of the equivalence classes of those self-dual codes that intersect
in a codimension 1 subspace. The eigenspaces of this self-adjoint linear
operator may be described in terms of a coding-theory analogue of the Siegel
-operator
Jacobi polynomials and harmonic weight enumerators of the first-order Reed--Muller codes and the extended Hamming codes
In the present paper, we give harmonic weight enumerators and Jacobi
polynomials for the first-order Reed--Muller codes and the extended Hamming
codes. As a corollary, we show the nonexistence of combinatorial -designs in
these codes.Comment: 9 page
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