Encoder Lipschitz integers: the Lipschitz integers that have the “division with small remainder” property

A quaternion, each component is in integers, is called a Lipschitz integer. A Lipschitz integer is called a primitive Lipschitz integer just if the greatest common divisor of its components is one. Complex-valued codes over Lipschitz integers are obtained accompanied by a respective modulo function. In this study, we demonstrate that some primitive Lipschitz integers, considered suitable for encoding, are inappropriate for constructing complex-valued codes over Lipschitz integers. To solve this problem, we investigate the primitive Lipschitz integers that have the “division with small remainder” property. We construct a named set “Encoder Lipschitz integers,” consists of primitive Lipschitz integers that have the “division with small remainder” property. In addition, we examine some algebraic properties of this set.