digital sequences? In algebraic terms, are there any other possibilities to add digit vectors than addition with or without carry? If yes, then this would lead to additional types of groups and dual groups and, hence, to additional function systems. If not, then the Walsh functions and the $b$-adic functions suffice to analyze digital sequences.

For details, in particular for the proofs in the following considerations and a refinement using compositions of positive integers instead of partitions and automorphisms of certain groups to define additions, we refer the reader to [@Hel12c].

Let $b\ge 2$ be a fixed integer and let $\mathcal{A}_b= \{ 0,1, \ldots, b-1\}$ denote the set of $b$-ary digits. For $m\in {\ensuremath{\mathbb{N}}}$, $m\ge 2$, let $\mathcal{A}_b^m$ stand for the $m$-fold cartesian product of the set $\mathcal{A}_b$ with itself. We study the following question: What are the binary operations “$+$” on the set $\mathcal{A}_b^m$ of digit vectors of length $m$ such that the pair $(\mathcal{A}_b^m, +)$ is an abelian group?

In this paper, when we speak of an “addition on $\mathcal{A}_b^m$”, we mean a binary operation “$+$” on the set $\mathcal{A}_b^m$ of digit vectors in base $b$ such that the pair $(\mathcal{A}_b^m, +)$ is an abelian group. The reader 