Remarks on Gibbs Permutation Matrices for Ternary Bent Functions

As in the binary case, ternary bent functions are a very small portion of the set of all ternary functions for a given number of variables. For example, for n = 2, there are 486 ternary bent functions out of 19683 ternary functions, which is 2, 47%, and this number reduces exponentially with the increase of n. However, finding, or alternatively, constructing them is a challenging task. A possible approach is based upon the manipulation of known ternary bent functions to construct other ternary bent functions. In this paper, we define Gibbs permutation matrices derived from the Gibbs derivative with respect to the Vilenkin-Chrestenson transform and propose their usage in constructing bent functions. The method can be extended to p-valued bent functions, where p is a prime larger than 3.