Robert Craigen

The associativity equation revisited

SummaryConsideration of the Associativity Equation,x ∘ (y ∘ z) = (x ∘ y) ∘ z, in the case where∘:I × I → I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( − ∞,b), ( − ∞,b], −∞, +∞), (a, + ∞), or [a, + ∞) — whereb = 0 or −1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.

Complex Golay sequences: structure and applications

Abstract Complex Golay sequences were introduced in 1992 to generalize constructions for Hadamard matrices using Golay sequences. (In the last section of this paper we describe some independent earlier work on quadriphase pairs–equivalent objects used in the setting of signal processing.) Since then we have constructed some new infinite classes of these sequences and learned some facts about their structure. In particular, if the length of complex Golay sequences is divisible by a prime p≡3 mod 4 , then their Hall polynomials have a nontrivial factorization h(x)k(x), cx d h(x)k ∗ (x) as polynomials over GF(p2), where c=a+bi, a 2 +b 2 ≡−1 mod p and k ∗ is obtained from k by a natural involution acting on complex Laurent polynomials. We explain how these facts can be used to simplify the search for complex Golay sequences, and show how to construct a large variety of sets of four complex sequences with zero autocorrelation, suitable for the construction of various matrices such as Hadamard matrices, complex Hadamard matrices and signed group Hadamard matrices over the dihedral signed group.

A Theory of Ternary Complementary Pairs

Sequences with zero autocorrelation are of interest because of their use in constructing orthogonal matrices and because of applications in signal processing, range finding devices, and spectroscopy. Golay sequences, which are pairs of binary sequences (i.e., all entries are ±1) with zero autocorrelation, have been studied extensively, yet are known only in lengths 2a10b26c. Ternary complementary pairs are pairs of (0, ±1)-sequences with zero autocorrelation (thus, Golay pairs are ternary complementary pairs with no 0s). Other kinds of pairs of sequences with zero autocorrelation, such as those admitting complex units for nonzero entries, are studied in similar contexts. Work on ternary complementary pairs is scattered throughout the combinatorics and engineering literature where the majority approach has been to classify pairs first by length and then by deficiency (the number of 0s in a pair); however, we adopt a more natural classification, first by weight (the number of nonzero entries) and then by length. We use this perspective to redevelop the basic theory of ternary complementary pairs, showing how to construct all known pairs from a handful of initial pairs we call primitive. We display all primitive pairs up to length 14, more than doubling the number that could be inferred from the existing literature.

Signed groups, sequences, and the asymptotic existence of Hadamard matrices

Abstract We use the newly developed theory of signed groups and some known sequences with zero autocorrelation to derive new results on the asymptotic existence of Hadamard matrices. New values of t are obtained such that, for any odd number p , there exists an Hadamard matrix of order 2 t p . These include: t = 2 N , where N is the number of nonzero digits in the binary expansion of p , and t = 4⌈ 1 6 log 2 ( (p − 1) 2 ⌉ + 2 . Both numbers improve on all previous general results, but neither uses the full power of our method. We also discuss some of the implications of our method in terms of signed group Hadamard matrices and signed group weighing matrices : There exists a circulant signed group Hadamard matrix of every even order n , using a suitable signed group. This result stands in striking contrast to the known results for Hadamard matrices and complex Hadamard matrices, and the circulant Hadamard matrix conjecture. Signed group weighing matrices of even order n always exist, with any specified weight w ⩽ n .

Further explorations into ternary complementary pairs

In [R. Craigen, C. Koukouvinos, A theory of ternary complementary pairs, J. Combin. Theory Ser. A 96 (2001) 358-375], we proposed a systematic approach to the theory of ternary complementary pairs (TCPs) and showed how all pairs known then could be constructed using a single elementary product, the natural equivalence relations, and a handful of pairs which we called primitive. We also introduced more new primitive pairs than could be inferred previously, concluding with some conjectures reflecting the patterns that were beginning to arise in light of the new approach.In this paper we take what appears to be the natural next step, by investigating these patterns among those lengths and weights that are within easy computational distance from the last length considered therein, length 14. We give complete results up to length 21, and partial results up to length 28. (Ironically, although we proceed analytically by weight first then length, for computational reasons we are bound, in this empirical investigation, to proceed according to length first.)Thus we provide support for the previous conjectures, and shed enough new light to speculate further as to the likely ultimate shape of the theory. Since short term work on TCPs will require massive acquisition of data about small pairs, we also discuss affixes--a computational strategy that arose out of the investigations culminating in this article.

The structure of weighing matrices having large weights

We examine the structure of weighing matricesW(n, w), wherew=n−2,n−3,n−4, obtaining analogues of some useful results known for the casen−1. In this setting we find some natural applications for the theory ofsigned groups and orthogonal matrices with entries from signed groups, as developed in [3]. We construct some new series of Hadamard matrices from weighing matrices, including the following:W(n, n−2) implies an Hadamard matrix of order2n ifn≡0 mod 4 and order 4n otherwise;W(n, n−3) implies an Hadamard matrix of order 8n; in certain cases,W(n, n−4) implies an Hadamard matrix of order 16n. We explicitly derive 117 new Hadamard matrices of order 2tp, p<4000, the smallest of which is of order 23·419.

Product of four Hadamard matrices

We prove that if there exist Hadamard matrices of order 4m, 4n, 4p, and 4q then there exists an Hadamard matrix of order 16mnpq. This improves and extends the known result of Agayan that there exists a Hadamard matrix of order 8mn if there exist Hadamard matrices of order 4m and 4n.

EQUIVALENCE CLASSES OF INVERSE ORTHOGONAL AND UNIT HADAMARD MATRICES

In 1867, Sylvester considered n × n matrices, ( a ij ), with nonzero complex-valued entries, which satisfy ( a ij )( a ij −1 ) = nI Such a matrix he called inverse orthogonal . If an inverse orthogonal matrix has all entries on the unit circle, it is a unit Hadamard matrix , and we have orthogonality in the usual sense. Any two inverse orthogonal (respectively, unit Hadamard) matrices are equivalent if one can be transformed into the other by a series of operations involving permutation of the rows and columns and multiplication of all the entries in any given row or column by a complex number (respectively a number on the unit circle). He stated without proof that there is exactly one equivalence class of inverse orthogonal matrices (and hence also of unit Hadamard matrices) in prime orders and that in general the number of equivalence classes is equal to the number of distinct factorisations of the order. In 1893 Hadamard showed this assertion to be false in the case of unit Hadamard matrices of non-prime order. We give the correct number of equivalence classes for each non-prime order, and orders ≤ 3, giving a complete, irredundant set of class representatives in each order ≤ 4 for both types of matrices.

Boolean and ternary complementary pairs

A ternary complementary pair, TCP(n, w), is a pair of (0, ± 1)-sequences of length n with zero autocorrelation and weight w. These are of theoretical interest in combinatorics as well as of practical consequence in coding, transmitting and processing various kinds of signals. When one attempts to construct a TCP of given length and weight, the first thing to decide is where to place the zeros, if any. Thus arise Boolean complementary pairs, BP(n, w)--pairs of (0, 1)- sequences of length n with zero autocorrelation over Z2 and a total of w 1s. The unique pair of (0, 1)-sequences having the same support as a TCP(n, w) is a BP(n, w) (but the converse is not necessarily true); thus, Boolean complementary pairs establish candidate zero patterns for ternary complementary pairs. This cleanly separates the construction of ternary complementary pairs into two stages: deciding where to put the zeros, and determining the sign of the nonzero entries. We obtain some necessary conditions for the existence of Boolean complementary pairs. We conduct an exhaustive survey of pairs of small lengths and construct some infinite classes clearly of fundamental importance in the theory. We completely characterize all pairs of even weight and give a product construction for pairs of odd weight that gives a greater variety of new pairs than similar product methods used in the ternary case.

Embedding rectangular matrices in hadamard matrices

It is shown that any m×n±1 matrix may be embedded in a Hadamard matrix of order kl, where k and l are the least orders greater than or equal to m and nrespectively in which Hadamard matrices exist.