A construction of unimodular equiangular tight frames from resolvable Steiner systems

Abstract

ABSTRACT An equiangular tight frame (ETF) is an M N matrix which has orthogonal equal norm rows, equal normcolumns, and the inner products of all pairs of columns have the same modulus. In this paper we studyETFs in which all of the entries are unimodular, and in particular pth roots of unity. A new constructionof unimodular ETFs based on resolvable Steiner systems is presented. This construction gives many newexamples of unimodular ETFs. In particular, an new in nite class of ETFs with entries in f1; 1g ispresented.Keywords: tight frames, equiangular, Hadamard matrices, Steiner systems, resolvable 1. INTRODUCTION Let F be an M N matrix with real or complex entries. If there exists A > 0 such that FF = AI , where Iis the identity matrix, then F is called a tight frame . If F F has constant diagonal and all of the o -diagonalentries have the same modulus, then F is called equiangular. If all of the entries of F have modulus 1 thenwe call F unimodular . If all of the entries of F are pth roots of unity, then we call F unimodular of degree p.This article is concerned with equiangular tight frames (ETFs) which are unimodular of degree p. Inparticular, we are concerned with nding unimodular ETFs of degree 2, which we shall also refer to asHadamard ETFs. Note that in the usual de nition of ETFs the columns are taken to have norm 1. However,since we are concerned with unimodular ETFs we will avoid normalizing, thus the columns of an M Nunimodular ETF have norm M .The paper is organized as follows. In Section 2 we present some necessary conditions on the existence ofunimodular ETFs. In Section 3 we review the construction of harmonic ETFs, which is an important classof ETFs that also happen to be unimodular. In Section 4 we review the construction of the Steiner ETFs ofFickus, Mixon, and Tremain,