Multiplication of ternary complementary pairs

We present a computer-search method for concatenating or multiplying binary or ternary complementary pairs. All multiplications by a particular number m are considered. The computer-search method is new and leads to a large set of new results. The results and equivalences are discussed and some applications and numerical consequences are shown

Algorithms for searching for normal and near- Yang sequences

This thesis is about normal and near-Yang sequences. An original definition of near-Yang sequences is included. After a short overview of the mathematical background of normal and near-Yang sequences, different algorithms for searching for these sequences are described. These algorithms can be divided in two groups: exhaustive search algorithms and heuristic search algorithms. One of the most important heuristic search algorithms is the simulated annealing algorithm. The following new results were found: Near-Yang sequences with weight 12 do exist for the following lengths ^ = 7,11,13,15. Normal sequences of length n = 24 do not exist. An exhaustive search for length n = 25 has been carried out for about 80% of the search-space and new normal sequences of length 25 have been found. The thesis concludes with a discussion of the algorithms and the results, and directions for further research are suggested. New results obtained from this research wiU appear in New Results with Near- Yang Sequences , Utilitas Mathematica, which is accepted for publication. Key words: Normal sequences, near-Yang sequences, autocorrelation function, exhaustive search algorithm, heuristic search algorithm, simulated annealing

Combinatorial designs, sequences and cryptography

A major part of this thesis is about sequences with low, zero or constant autocorrelation function and their implications to combinatorial designs and cryptography. We discuss sequences with zero (periodic or nonperiodic) autocorrelation function and we show an array of constructions, also called multiplications, that give many new infinite families of such sequences. We introduce cyclotomy and show how such sequences together with a search through incidence matrices of cyclotomic or generalised cosets can be used to find a variety of new combinatorial designs. These designs include new weighing matrices, orthogonal designs and D-optimal designs. In particular we give: • T-matrices of order n for n = 13,19,31,37,41,42,43,61,66,86,87. • Weighing matrices W(4n, 4n - 2) and W(4n, 2n - 1) for n = 25,31,37,41,61,71,73,157. • D-optimal designs of order v = 2 = 31,33,37,41,43,61,73,85,91,93,113,145,157,181. We review some of the Stanton-Spratt-Whiteman constructions for difference sets and show how they can be applied to construct supplementary difference sets. We also give many other infinite families of supplementary difference sets and pairwise balanced designs with .A = 1. The thesis concludes with some discussions about low autocorrelation in cryptography and a presentation of a small one-key cryptosystem

On 4-NPAF(1,2w) sequences

We give an overview on the existence of 4-NPAF (1,2w) sequences. We sketch some construction methods which give new 4-NPAF(1,2w) sequences and new orthogonal designs OD(4n;1,2w)

The weighing matrices of order 4n and weight 4n-2 and 2n-1

We give algorithms and constructions for mathematical and computer searches which allow us to establish the existence of W(4n, 4n - 2) and W (4n, 2n - 1) for many orders 4n less than 4000. We compare these results with the orders for which W(4n, 4n) and W(4n, 2n) are known. We use new algorithms based on the theory of cyclotomy to obtain new T-matrices of order 43 and JM-matrices which yield W(4n, 4n - 2) for n = 5,7,9,11,13,17,19,25,31,37,41,43,61,71,73,157

New results with near- Yang sequences

We construct new TW -sequences, weighing matrices and orthogonal designs using near-Yang sequences. In particular we construct new OD(60(2m + 1) + 4t; 13(2m+ 1), 13(2m+ 1), 13(2m+ 1), 13(2m+ 1) and new W(60(2m+ 1) + 4t; 13s(2m+ 1))for all t ≥ O, m ≤ 30, s = 1,2,3,4

New normal sequences of length 25

An introduction to binary sequences, combinatorial designs and how they are related to communication theory and computer security is given. An exhaustive search algorithm for normal sequences is presented. This is the first time that the lengths n = 24 and n = 25 have been searched through completely. No sequences of length 24 are found. It turns out that all the normal sequences of length 25 can be derived from Turyn sequences. This construction is subject to a new theorem that is given here

On new families of supplementary difference sets over rings with short orbits

... We also show some new balanced incomplete block designs and pairwise balanced designs arising from the constructions given here