Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property

Considering error-correcting codes over Hurwitz integers, prime Hurwitz integers are considered. On the other hand, considering transmission over Gaussian channel, Hurwitz integers, whose the norm is either a prime integer or not a prime integer, are considered. In this study, we consider Hurwitz integers, the greatest common divisor of components of which is one, i.e., primitive Hurwitz integers. We show, with the help of a proposition, that some primitive Hurwitz integers accompanied by a related modulo function are not suitable for constructing Hurwitz signal constellations. To solve this problem, we show, with the help of a proposition, the existence of primitive Hurwitz integers that have the "division with small remainder" property used to construct the Hurwitz constellations. We also call the set of these integers named as "Encoder Hurwitz Integers" set. Moreover, we examine some properties of the mentioned set. In addition, we investigate the performances of Hurwitz signal constellations, which are constructed accompanied by a related modulo function using Hurwitz integers, each component of which is in half-integers, for transmission over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).