Finite field and group algorithms for orthogonal sequence search

Abstract

Introduction: Hadamard matrices consisting of elements 1 and –1 are an ideal object for a visual application of finite dimensional mathematics operating with a finite number of addresses for –1 elements. The notation systems of abstract algebra methods, in contrast to the conventional matrix algebra, have been changing intensively, without being widely spread, leading to the necessity to revise and systematize the accumulated experience. Purpose: To describe the algorithms of finite fields and groups in a uniform notation in order to facilitate the perception of the extensive knowledge necessary for finding orthogonal and suborthogonal sequences. Results: Formulas have been proposed for calculating relatively unknown algorithms (or their versions) developed by Scarpis, Singer, Szekeres, Goethal — Seidel, and Noboru Ito, as well as polynomial equations used to prove the theorems about the existence of finite-dimensional solutions. This replenished the significant lack of information both in the domestic literature (most of these issues are published here for the first time) and abroad. Practical relevance: Orthogonal sequences and methods for their effective finding via the theory of finite fields and groups are of direct practical importance for noise-immune coding, compression and masking of video data.