Unitary designs and codes

Details Unitary designs and codes Unitary designs and codes

2008-09-22

arxiv.org/abs/0809.3813v1

Des. Codes Cryptogr. 53, 13-31 (2009)

Mathematics

Combinatorics

25 pages, no figures

Scientific paper

10.1007/s10623-009-9290-2

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code - a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values - and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.

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