Kirkman Equiangular Tight Frames and Codes
IEEE Transactions on Information Theory(2014)
摘要
n equiangular tight frame (ETF) is a set of unit vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design and compressed sensing. At the moment, there are only two known flexible methods for constructing ETFs: harmonic ETFs are formed by carefully extracting rows from a discrete Fourier transform; Steiner ETFs arise from a tensor-like combination of a combinatorial design and a regular simplex. These two classes seem very different: the vectors in harmonic ETFs have constant amplitude, whereas Steiner ETFs are extremely sparse. We show that they are actually intimately connected: a large class of Steiner ETFs can be unitarily transformed into constant-amplitude frames, dubbed Kirkman ETFs. Moreover, we show that an important class of harmonic ETFs is a subset of an important class of Kirkman ETFs. This connection informs the discussion of both types of frames: some Steiner ETFs can be transformed into constant-amplitude waveforms making them more useful in waveform design; some harmonic ETFs have low spark, making them less desirable for compressed sensing. We conclude by showing that real-valued constant-amplitude ETFs are equivalent to binary codes that achieve the Grey–Rankin bound, and then construct such codes using Kirkman ETFs.
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关键词
tensor-like combination,equiangular tight frame,welch bound equality sequence,kirkman equiangular tight frame,euclidean space,constant-amplitude waveforms,harmonic etf,welch-bound-equality sequences,codes,compressed sensing,waveform design,steiner etf,welch bound,real-valued constant-amplitude etf,grey-rankin bound
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