Kirkman Equiangular Tight Frames and Codes

TL;DR

This paper reveals a deep connection between two classes of equiangular tight frames, harmonic and Steiner ETFs, and introduces Kirkman ETFs, which unify their properties, with implications for waveform design and coding theory.

Contribution

The paper demonstrates that many Steiner ETFs can be transformed into constant-amplitude frames called Kirkman ETFs and shows that harmonic ETFs are a subset of Kirkman ETFs, unifying their frameworks.

Findings

Many Steiner ETFs can be unitarily transformed into constant-amplitude frames.

Harmonic ETFs are a subset of Kirkman ETFs.

Real-valued constant-amplitude ETFs correspond to binary codes achieving the Grey-Rankin bound.

Abstract

An 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.