A Note on Vectorial Boolean Functions as Embeddings

Let F be a vectorial Boolean function from Fⁿ to Fᵐ, where m ≥ 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 2ᵐ - 2ᵐ⁻ⁿ components of F can be balanced, and this maximum is achieved precisely when F is an embedding, with the remaining 2ᵐ⁻ⁿ components being constants. Additionally, for quadratic embeddings, we demonstrate that there are always at least 2ⁿ - 1 balanced components when n is even, and 2ᵐ⁻¹ + 2ⁿ⁻¹ - 1 balanced components when n is odd.