An introduction to cocyclic generalised Hadamard matrices

• Published in: Discrete Applied Mathematics 2000-05, Vol.102 (1), p.115-131
• Main Author: Horadam, K.J.
• Format: Article
• Language: English
• Subjects: Combinatorics; Combinatorics. Ordered structures; Designs and configurations; Exact sciences and technology; Generalised Hadamard matrix; Group theory; Group theory and generalizations; Mathematics; Orthogonal cocycle; Sciences and techniques of general use; Semiregular relative difference set
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/S0166-218X(99)00233-4

Many codes and sequences designed for robust or secure communications are built from Hadamard matrices or from related difference sets or symmetric block designs. If an alphabet larger than {0,1} is required, the natural extension is to generalised Hadamard matrices, with entries in a group. The code and sequence construction techniques for Hadamard matrices are applicable to the general case. A cocyclic generalised Hadamard matrix with entries in an abelian group is equivalent to a semiregular central relative difference set and to a divisible design with a regular group of automorphisms, class regular with respect to the forbidden central subgroup. In this introduction we outline the necessary background on cocycles and their properties, give some familiar examples of this unfamiliar concept and demonstrate the equivalence of the above-mentioned objects. We present recent results on the theory of cocyclic generalised Hadamard matrices and their applications in one area: error-correcting codes.