An Algebraic Construction Technique for Codes over Hurwitz Integers

Abstract

Let α be a prime Hurwitz integer. Hα, 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 modulo α and, the domain of related modulo function in this study is ZN(α), which is residual class ring of ordinary integers with N(α) elements, which is the norm of prime Hurwitz integer α. 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(α) 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 presented 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.