On the dimension of an APN code

Abstract

A map f : V: = GF(2^m) → V is APN (almost perfect nonlinear) if its directional derivatives in nonzero directions are all 2-to-1. If m is greater than 2 and f vanishes at 0, then this derivative condition is equivalent to the condition that the binary linear code of length 2^m − 1, whose parity check matrix has jth column equal to ${\omega^j \brack f(\omega^j)}$, is double-error-correcting, where ω is primitive in V. Carlet et al. (Designs Codes Cryptogr 15:125–156, 1998) proved that this code has dimension 2^m − 1 − 2m; but their indirect proof uses a subtle argument involving general code parameter bounds to show that a double-error correcting code of this length could not be larger. We show here that this result follows immediately from a well-known result on bent functions ...a subject dear to the heart of Jacques Wolfmann.