A Note on Vectorial Boolean Functions as Embeddings

Let $F$ be a vectorial Boolean function from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, with $m \geq n$. We define $F$ as an embedding if $F$ is injective. In this paper, we examine the component functions of $F$, focusing on constant and balanced components. Our findings reveal that at most $2^m - 2^{m-n}$ components of $F$ can be balanced, and this maximum is achieved precisely when $F$ is an embedding, with the remaining $2^{m-n}$ components being constants. Additionally, for partially-bent embeddings, we demonstrate that there are always at least $2^n - 1$ balanced components when $n$ is even, and $2^{m-1} + 2^{n-1} - 1$ balanced components when $n$ is odd. A relation with APN functions is shown.