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

A function \(f: F_{2^m} \rightarrow F_{2^t}\) 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 \(Tr(\lambda x^d)\) (for \(d = 2^{l} + 1, 2^{2l} - 2^{l} + 1\) and \(\lambda \in F_{2^m}^{*}\) ). 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.