Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Codes and Siegel modular forms
Discrete Mathematics
A Family of Optimal Packings in Grassmannian Manifolds
Journal of Algebraic Combinatorics: An International Journal
A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces
Journal of Algebraic Combinatorics: An International Journal
Spherical 7-Designs in 2^n-Dimensional Euclidean Space
Journal of Algebraic Combinatorics: An International Journal
Quantum error correction via codes over GF(4)
IEEE Transactions on Information Theory
Type II codes, even unimodular lattices, and invariant rings
IEEE Transactions on Information Theory
Jacobi forms over totally real fields and type II codes over Galois rings GR(2m, f)
European Journal of Combinatorics - Special issue on arithmétique et combinatoire
Type II Codes over$$\mathbb{Z}/2k\mathbb{Z}$$, Invariant Rings and Theta Series
Designs, Codes and Cryptography
Matrix Groups Related to the Quaternion Group and Spherical Orbit Codes
Designs, Codes and Cryptography
Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond
Quantum Information & Computation
Complete weight enumerators of generalized doubly-even self-dual codes
Finite Fields and Their Applications
A linearized stabilizer formalism for systems of finite dimension
Quantum Information & Computation
Magic-state distillation with the four-qubit code
Quantum Information & Computation
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The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m(m \neq 3 ) is a subgroup of index 2 in a certain “Clifford group” \mathcal{C}_m of structure 2_+^{1+2m} . O^+(2m,2). This group and its complex analogue \mathcal{X}_m of structure (2_+^{1+2m}{\sf Y}Z_8) . Sp(2m, 2) 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@apos;s 1996 result that the space of invariants for \mathcal{C}_m of degree 2k is spanned by the complete weight enumerators of the codes C \otimes \Bbb{F}_{2^m}, where C ranges over all binary self-dual codes of length 2k; these are a basis if m \ge k-1. We also give new constructions for L_m and \mathcal{C}_m: let M be the \Bbb{Z}[\sqrt{2}]-lattice with Gram matrix \scriptsize\big[\begin{array}{@{}r@{\quad}r@{}} 2 & \sqrt{2} \\ \sqrt{2} & 2 \end{array} \big]. Then L_m is the rational part of M^{\otimes m}, and \mathcal{C}_m = {{\rm Aut}} (M^{\otimes m} ). Also, if C is a binary self-dual code not generated by vectors of weight 2, then \mathcal{C}_m is precisely the automorphism group of the complete weight enumerator of C \otimes \Bbb{F}_{2^m}. There are analogues of all these results for the complex group \mathcal{X}_m, with “doubly-even self-dual code” instead of “self-dual code.”