Codes, Graphs and Schemes from Nonlinear Functions

AMS classifications: 05E30; 05B20; 94B0

Codes, graphs and schemes from nonlinear functions

AbstractWe consider functions on binary vector spaces which are far from linear functions in different senses. We compare three existing notions: almost perfect nonlinear functions, almost bent (AB) functions, and crooked (CR) functions. Such functions are of importance in cryptography because of their resistance to linear and differential attacks on certain cryptosystems. We give a new combinatorial characterization of AB functions in terms of the number of solutions to a certain system of equations, and a characterization of CR functions in terms of the Fourier transform. We also show how these functions can be used to construct several combinatorial structures; such as semi-biplanes, difference sets, distance regular graphs, symmetric association schemes, and uniformly packed (BCH and Preparata) codes

On the Weight Distribution of Codes over Finite Rings

Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f for which the homogeneous weight distribution of C can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.Comment: 18 p