Christos Koukouvinos

New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function - a review

Abstract The book, Orthogonal Designs : Quadratic Forms and Hadamard Matrices , Marcel Dekker, New York-Basel, 1979, by A.V. Geramita and Jennifer Seberry, has now been out of print for almost two decades. Many of the results on weighing matrices presented therein have been greatly improved. Here we review the theory, restate some results which are no longer available and expand on the existence of many new weighing matrices and orthogonal designs of order 2 n where n is odd. We give a number of new constructions for orthogonal designs. Then using number theory, linear algebra and computer searches we find new weighing matrices and orthogonal designs. We have reviewed completely the weighing matrix conjecture for orders 2n, n⩽35, n odd. The previously known results for weighing matrices for n ⩽21 are summarized here, and new results given, leaving three unresolved cases. The results for weighing matrices for n ⩾23 are presented here for the first time. For orders n, 23⩽n⩽25, 3 remain unsolved as do a further 106 cases for orders 27⩽ n ⩽49. We also review completely the orthogonal design conjecture for two variables in orders ≡2 ( mod 4) . The results for orders 2n, n odd, 15⩽ n ⩽33 being given here for the first time.

On self-dual codes over some prime fields

In this paper, we study self-dual codes over GF(p) where p = 11, 13, 17, 19, 23 and 29. A classification of such codes for small lengths is given. The largest minimum weights of these codes are investigated. Many maximum distance separable self-dual codes are constructed.

Hadamard matrices, orthogonal designs and construction algorithms

We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction using Williamson matrices, Legendre pairs and the discret Fourier transform and the two circulants construction.Next we move to algorithms to determine the equivalence of Hadamard matrices using the profile and projections of Hadamard matrices. A summary is then given which considers inequivalence of Hadamard matrices of orders up to 44.The final two sections give algorithms for constructing orthogonal designs, short amicable and amicable sets for use in the Kharaghani array.

Weighing matrices and their applications

Abstract Three major applications of weighing matrices are discussed. New weighing matrices and skew weighing matrices are given for many orders 4t ⩽ 100. We resolve the skew-weighing matrix conjecture in the affirmative for 4t ⩽ 88.

Supplementary difference sets and optimal designs

Abstract D-optimal designs of order n = 2 v ≡ 2 (mod 4), where q is a prime power and v = q2 + q + 1 are constructed using two methods, one with supplementary difference sets and the other using projective planes more directly. An infinite family of Hadamard matrices of order n = 4v with maximum excess σ(n) = n n−3 where q is a prime power and v = q2 + q + 1 is a prime, is also constructed.

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.

Hadamard ideals and Hadamard matrices with two circulant cores

We apply computational algebra methods to the construction of Hadamard matrices with two circulant cores, given by Fletcher, Gysin and Seberry. We introduce the concept of Hadamard ideal, to systematize the application of computational algebra methods for this construction. We use the Hadamard ideal formalism to perform exhaustive search constructions of Hadamard matrices with two circulant cores for the twelve orders 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52. The total number of such Hadamard matrices is proportional to the square of the parameter. We use the Hadamard ideal formalism to compute the proportionality constants for the twelve orders listed above. Finally, we use the Hadamard ideal formalism to improve the lower bounds for the number of inequivalent Hadamard matrices for the seven orders 44, 48, 52, 56, 60, 64, 68.

Exploring k-circulant supersaturated designs via genetic algorithms

E(s^2)-optimal or near optimal, two level k-circulant supersaturated designs are explored by means of genetic algorithms. All known k-circulant classes for k=2,...,6 have been rebuilt and improved. The successful application of genetic algorithms is further illustrated by the construction of several k-circulant supersaturated designs for k=7,8.

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.

On skew-Hadamard matrices

Skew-Hadamard matrices are of special interest due to their use, among others, in constructing orthogonal designs. In this paper, we give a survey on the existence and equivalence of skew-Hadamard matrices. In addition, we present some new skew-Hadamard matrices of order 52 and improve the known lower bound on the number of the skew-Hadamard matrices of this order.