Bent and Near-Bent Function Construction and 2-Error-Correcting Codes

A function f : F2(m) -> F-2t is called a vectorial Boolean function in m variables. Whenever t = 1 we call these functions Boolean functions. In this work, we study the construction of Gold and Kasami-Welch functions of the form T r (lambda(d)(x)) (for d = 2(l) + 1, 2(2l)- 2(l) + 1 and lambda is an element of F-2m* These functions' nonlinearity property is a measure of their distance to the set of affine functions (the first-order Reed-Muller codes). We generalize a result of Dillon and Dobbertin for conditions under which these functions are bent. We give algorithms that generate and determine the bentness of the functions. We construct 2-error-correcting cyclic codes utilizing Almost-Perfect-Nonlinear (APN) and near-bent exponents. We present theorems that enumerate the Gold and Kasami-Welch functions. We improve previous algorithms used to determine the minimum distance of these codes.