Optimal Representations of Gaussian and Eisenstein Integers using digit sets closed under multiplication

TL;DR

This paper investigates optimal digit representations of Gaussian and Eisenstein integers in specialized numeration systems, providing bounds and characterizations for minimal non-zero digit representations, with implications for efficient complex number computations.

Contribution

It introduces bounds and characterizations for optimal representations of Gaussian and Eisenstein integers in two specific numeration systems, extending previous binary system results.

Findings

Upper bounds on the number of optimal representations depending on the 3-NAF digits

Characterization of Gaussian integers attaining the upper bound

Every Eisenstein integer has an optimal 2-NAF representation

Abstract

We study two positional numeration systems which are known for allowing very efficient addition and multiplication of complex numbers. The first one uses the base $\beta = \imath - 1$ and the digit set $\mathcal{D} = \{ 0, \pm 1, \pm \imath \}$. In this numeration system, every non-zero Gaussian integer~$x$ has an infinite number of representations. We focus on optimal representations of~$x$ -- i.e., representations with minimal possible number of non-zero digits. One of the optimal representations of~$x$ has the so-called $3$-non-adjacent form ($3$-NAF). We provide an upper bound on the number of distinct optimal representations of~$x$, depending on the number of non-zero digits in the $3$-NAF of~$x$. We also characterize the Gaussian integers for which the upper bound is attained. The same questions are answered also for the second numeration system with base $\beta = \omega - 1$ and digit set $\mathcal{D} = \{ 0, \pm 1, \pm \omega, \pm \omega^2 \}$, where $\omega = \exp(2\pi\imath / 3)$. In this system, every Eisenstein integer has a $2$-NAF, which is optimal. This paper can be understood as an analogy to the result of Grabner and Heuberger obtained for the signed binary numeration system, using base $\beta = 2$ and digit set $\mathcal{D} = \{0, \pm 1\}$.