A note on vectorial boolean functions as embeddings

Seminario del Dipartimento di Matematica
23 May 2024
Start time 
5:40 pm
Polo Ferrari 1 - Via Sommarive 5, Povo (Trento)
Aula A218
Dipartimento di Matematica
Target audience: 
University community
UniTrento students
Contact person: 
Prof. Willem Adriaan De Graaf, dott.ssa Mima Stanojkovski
Contact details: 
Staff Dipartimento di Matematica
Augustine Musukwa (University of Trento/Mzuzu University)


Let f be any vectorial Boolean function from F^n to F^m, with m > n. We say that f is an embedding if f is injective. In this note we want to understand the components of f, especially constant components and balanced components. We determine that at most 2^m - 2^(m−n) components of f can be balanced, while this maximum is achieved precisely when f is an embedding and the other 2^(m−n) components are constants. If f is a quadratic embedding, then we also show that there is always at least 2^n − 1 balanced components if n is even and 2^(m−1) + 2^(n−1) − 1 balanced components if n is odd.