单一设计和代码
Unitary designs and codes
摘要：
酉设计是近似整个酉群的酉矩阵的集合，非常类似于近似整个单位球面的球面设计。本文利用酉群的不可约表示，对任意d和t，给出了U（d）中酉t-设计的规模的一般下界.我们还引入了酉码的概念--U（d）的一个子集，其中任意一对矩阵的迹内积仅限于少数不同的值--并给出了对于任意d和s，U（d）中s次码的大小的上界。当已知代码或设计中出现的特定内积值时，可以加强这些界限。最后，我们描述了一些设计的构造：我们给出了U（d）中最小加权酉t-设计的大小的上界，并对有限群中的一些t-设计进行了分类。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
