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We give a construction of bent functions in dimension 2m from near-bent functions in dimension 2m−1. In particular, we give the first ever examples of non-weakly-normal bent functions in dimensions 10 and 12, which demonstrates the significance of our construction.

In this article a technique for constructing $p$-ary bent functions from near-bent functions is presented. Two classes of quadratic $p$-ary functions are shown to be near-bent. Applying the construction of bent functions to these classes of near-bent functions yields classes of non-quadratic bent functions. We show that one construction in even dimension yields weakly regular bent functions. For other constructions, we obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions.

We present an alternative proof of a result of Zeng–Shan–Hu that shows that the cyclic code with three zeros α, α 3 , α 13 has the same weight distribution as the 3-error-correcting BCH code. Our proof uses the theory of algebraic curves over finite fields, and combines results that are already in the literature. This method is applicable in other cases too.