## Date of Award

Winter 1998

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mathematics & Statistics

## Program/Concentration

Computational and Applied Mathematics

## Committee Director

Charlie H. Cooke

## Committee Member

John M. Dorrepaal

## Committee Member

Hideaki Kaneko

## Committee Member

Linda L. Vahala

## Abstract

Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology.

The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is *q* and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices *BH*(*p, q*)”, first discovered by A. T. Butson [6].

In this dissertation it is shown that an error correcting code whose codewords consist of real numbers in finite Galois field *Gf*(* p*) can be associated in a simple way with each Butson Hadamard matrix *BH*(*p, q*), where *p* > 0 is a prime number. Distance properties of such codes are studied, as well as conditions for the existence of linear codes, for which standard decoding techniques are available.

In the search for cyclic linear generalized Hadamard codes, the concept of an M-invariant infinite sequence whose elements are integers in a finite field is introduced. Such sequences are periodic of least period, *T*, and have the interesting property, that arbitrary identical rearrangements of the elements in each period yields a periodic sequence with the same least period. A theorem characterizing such M-invariant sequences leads to discovery of a simple and efficient polynomial method for constructing generalized Hadamard matrices whose core is a linear cyclic matrix and whose row vectors constitute a linear cyclic error correcting code.

In addition, the problem is considered of determining parameter sequences {*tn*} for which the corresponding potential generalized Hadamard matrices *BH*(*p, ptn*) do not exist. By analyzing quadratic Diophantine equations, new methods for constructing such parameter sequences are obtained. These results show the rich number theoretic complexity of the existence question for generalized Hadamard matrices.

## Rights

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).

## DOI

10.25777/06n5-9c93

## ISBN

9780599208780

## Recommended Citation

Heng, Iem H..
"Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups"
(1998). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/06n5-9c93

https://digitalcommons.odu.edu/mathstat_etds/82