An Algebraic Construction Technique for Codes over Hurwitz Integers

TL;DR

This paper introduces an algebraic construction method for codes over Hurwitz integers using a novel modulo function based on two modulo operations, and visualizes the residual class sets with various graph layout techniques.

Contribution

It presents a new algebraic construction technique for Hurwitz integer codes utilizing a dual modulo function approach, expanding the mathematical framework for such codes.

Findings

Residue class rings of Hurwitz integers with size N(α) are obtained.

Graphs of residual class sets are visualized using spring, high-dimensional, and spiral layouts.

Mathematical properties of the two modulo functions are analyzed.

Abstract

Let {\alpha} be a prime Hurwitz integer. H{\alpha}, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of H, which is the set of all Hurwitz integers. We consider left congruent module {\alpha} and, the domain of related modulo function in this study is ZN({\alpha}), which is residual class ring of ordinary integers with N({\alpha}) elements, which is the norm of prime Hurwitz integer {\alpha}. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. Thereby, we obtain the residue class rings of Hurwitz integers with N({\alpha}) size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we present graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.