Direct sums of balanced functions, perfect nonlinear functions, and orthogonal cocycles
Alain LeBel, Kathy J. Horadam · Journal of Combinatorial Designs · 2008
Abstract Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non‐cyclic abelian groups and use it to find all the orthogonal cocycles over Z2t, 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z2t, t ≥ 2, must be multiplicative. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 173–181, 2008