On Algebraic Properties of Primitive Eisenstein Integers with Applications in Coding Theory

An even Eisenstein integer is a multiple of an Eisenstein prime of the least norm. Otherwise, an Eisenstein integer is called odd. An Eisenstein integer that is not an integer multiple of another one is said to be primitive. Such integers can be used to construct signal constellations and complex-valued codes over Eisenstein integers via a carefully designed modulo function. In this work, we establish algebraic properties of even, odd, and primitive Eisenstein integers. We investigate conditions for the set of all units in a given quotient ring of Eisenstein integers to form a cyclic group. We perform set partitioning based on the multiplicative group of the set. This generalizes the known partitioning of size a prime number congruent to 1 modulo 3 based on the multiplicative group of the Eisenstein field in the literature.