Encoder Hurwitz Integers: The Hurwitz integers that have the ”division with small remainder” property

Abstract The modulo function is used to construct signal constellations over high-dimensional vector spaces such as Gaussian integers, Einstein-Jacobi integers, and quaternion integers. There is a one-to-one relationship between the Euclid division and the modulo function. If the Euclid division works for any quaternion integers, then it has the ”division with small remainder” property. The Hurwitz integers form a subring of ring of quaternions. Hurwitz constellations are constructed by using modulo with primitive Hurwitz integers whose norm is a prime integer. If the norm of Hurwitz integers is not a prime integer, then we can not set up an isomorphism between the residual class ring of ordinary integers and the left (right) equivalence class of Hurwitz integers since the size of sets is to be different. In this study, to solve this problem, we define a new set, named encoder Hurwitz integers, consisting of primitive Hurwitz integers that have the ”division with small remainder” property as an alternative instead of the set of primitive Hurwitz integers. Also, we investigate their performance over additive Gaussian noise (AWGN) channel by means of constellation figure of merit (CFM), average energy, minimum square Euclidean distance, and signal-to-noise ratio (SNR).