An algebraic construction technique for codes over Hurwitz integers

Authors : Ramazan DURAN, Murat GÜZELTEPE

Pages : 652-672

Doi:10.15672/hujms.1137425

View : 131 | Download : 221

Publication Date : 2023-05-30

Article Type : Research Paper

Abstract :Let $\\alpha$ be a prime Hurwitz integer. $\\mathcal{H}_{\\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\\mathcal{H},$ which is the set of all Hurwitz integers. 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. We consider left congruent modulo $\\alpha,$ and the domain of related modulo function is $\\mathbb{Z}_{Ninsert ignore into journalissuearticles values(\\alpha);},$ which is residual class ring of ordinary integers with $Ninsert ignore into journalissuearticles values(\\alpha);$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $Ninsert ignore into journalissuearticles values(\\alpha);$ 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 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.
Keywords : algebraic construction, average energy, block code, code rate, graph