具有少量不同内积和低相干性的群框架
Group Frames with Few Distinct Inner Products and Low Coherence
摘要：
近年来，框架理论一直是结构化信号和编码设计中的热门主题，其应用范围从压缩感知中的测量矩阵的设计到用于数据压缩和数据传输的球形码，到用于MIMO通信的空时码，以及量子感知中的测量算子。高性能代码通常产生于设计帧，其元素具有相互低的一致性。建立了原来的“组框架”的设计Slepian已经阐述了淡水河谷和沃尔德伦的作品，我们提出了几个新的框架结构的基础上循环和广义二面角群。Slepian最初的构造是基于这样的前提：群结构允许人们将具有$n$元素的帧中不同的内部成对内积的数量从$\frac{n(n-1)}{2}$减少到$n-1$。我们所有的构造都进一步利用群结构来产生紧框架，在框架元素之间具有更少的不同内积值。例如，当$n$是素数时，我们使用循环群来构造至多具有$\frac{n-1}{m}$不同内积的$m$维框架向量。我们使用这种行为来约束我们的框架的连贯性，通过参数的基础上的框架潜力，并得出更严格的界限，从组合和代数参数单独使用组结构。在某些情况下，我们恢复著名的韦尔奇界实现帧。在韦尔奇界尚未实现的情况下，并且不知道是可以实现的，我们得到的帧接近韦尔奇界的性能。
英文摘要：
Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. High-performance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original "group frame" design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian's original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with $n$ elements from $\frac{n(n-1)}{2}$ to $n-1$. All of our constructions further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When $n$ is prime, for example, we use cyclic groups to construct $m$-dimensional frame vectors with at most $\frac{n-1}{m}$ distinct inner products. We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance.
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