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

Abstract

The residue class set of a Hurwitz integer is constructed by modulo function with primitive Hurwitz integer whose norm is a prime integer, i.e. prime Hurwitz integer. In this study, we consider primitive Hurwitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Hurwitz integer is less than the norm of the primitive Hurwitz integer used to construct the residue class set of the Hurwitz integer, then, the Euclid division algorithm works for this primitive Hurwitz integer. The Euclid division algorithm always works for prime Hurwitz integers. In other words, the prime Hurwitz integers and halves-integer primitive Hurwitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Hurwitz residue class set that lies on primitive Hurwitz integers that their norms are not a prime integer and their components are in integers set. In this study, we solve this problem by defining Hurwitz integers that have the ”division with small remainder” property, namely, encoder Hurwitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Especially, Euclidean metric. Also, we investigate the performances of Hurwitz signal constellations (the left residue class set) obtained by modulo function with Hurwitz integers, which have the ”division with small remainder” property, 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).