Revisiting Butson Hadamard Matrix Construction Based on a Prominent Class of Pseudo-Random Sequences Yuri L. Borissov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia, 1113, Bulgaria +359-2-979-2832 youri@math.bas.bg Moon Ho Lee Institute of Information and Communication Chonbuk National University Jeonju, 561-756, R. Korea +82-63-270-2463 moonho@jbnu.ac.kr ABSTRACT We revisit the well known matrix construction method based on one of the most popular classes of p  ary pseudo-random sequences. Our detailed considerations explicitly show that both the ideal autocorrelation property and the balancedness are required to construct normalized Butson Hadamard matrices through that method. The refinement made is illustrated by some examples. Categories and Subject Descriptors B.4.1, C.2.0, C.2.1, E.4, F.2.1, G.1.3 General Terms Algorithms, Design, Theory, Verification. Keywords balancedness; Butson-type Hadamard matrix; ideal two-level autocorrelation. 1. INTRODUCTION Real and complex Hadamard matrices play an important role in many branches of mathematics and physics, as well as they have found numerous applications in contemporary technologies, such as, mass spectroscopy, polymer chemistry, signal and information processing including mobile, wireless and optical communication systems, geophysics, acoustics, nuclear medicine and nuclear physics, etc. (see, e.g. [Horadam 2007, Chapter 3] for more details). The first complex Hadamard matrices were found by Sylvester [1867], where he had observed that a special Vandermonde matrix based on p  th roots of unity multiplied by the constant 1 p is what nowadays we call unitary matrix, for any prime p . Soon after the publication of [Hadamard 1893] the interest was mainly on real Hadamard matrices, while the Sylvester's contribution fell into oblivion. Latter on, the interest to complex Hadamard matrices has been renewed by the works of Butson [1962] and Turyn [1970]. In particular, a class of complex Hadamard matrices all of whose entries are q -th roots of unity have been introduced and are known as Butson-type Hadamard (shortly BH) matrices [Butson 1962]. Considerable amount of efforts has also been devoted to find various constructions of real and complex Hadamard matrices and their modifications, generalizations and parameterizations. For a thorough surveys on this topic we refer to [Seberry and Yamada 1992] and [Yarlagadda and Hershey 1997]. In particular, the matrices of the most popular discrete transforms belong to this family and are very useful in practical applications. Here, we mention some of them: discrete Fourier transform (DFT) and its related (e.g., the discrete cosine and sine transforms), Walsh- Hadamard transform (WHT), central weighted Hadamard transform (CWHT) [Lee 1989], etc. Recent contributions to that field of research are [Matsufuji and Fan 2009] and [Szollosi 2007]. It is well known that sequence design could benefit from appropriate constructions of Hadamard matrices and vice versa (see, e.g., [Seberry and Yamada 1992]). In some sequence applications another aspects of this benefit may appear. Namely, if the transform associated with a given kind of sequences turns out to be equivalent to a transform having fast algorithm, this will help to design a fast algorithm for the former transform (see, e.g., [Cohn and Lempel 1977]). In the present paper, the general construction method of BH matrices using balanced complex sequences with ideal two-level autocorrelation property (see, e.g. [Jang et al. 2007], [Borissov et al. 2007]) is revisited. The origins of this method could be found in [McWilliams and Sloane 1976, Property-X, p. 1719], where a method for constructing classical real Hadamard matrices from binary m-sequences was given. The paper is organized as follows. In Sect. 2, we recall some definitions and properties. In Sect. 3, we give an description of the construction under consideration, and prove that it provides a normalized BH matrix. In Sect. 4, we illustrate this construction by examples, and finally, in Sect. 5 we end with some conclusions. 2. PRELIMINARIES In order to present our results we need to recall several definitions. DEFINITION 2.1: Let ( ) kl h  H be a square matrix of size n whose entries are complex numbers of magnitude 1. Denote by † H the transpose matrix of the matrix of complex conjugates of Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. CUBE 2012, September 3–5, 2012, Pune, Maharashtra, India. Copyright 2012 ACM 978-1-4503-1185-4/12/09…$10.00. 11