Necessary Conditions for the Existence of Regular $p$ -Ary Bent Functions

We find some necessary conditions for the existence of regular p-ary bent functions (from Zⁿp to Zp), where p is a prime. In more detail, we show that there is no regular p-ary bent function f in n variables with w(M_f) larger than n/2, and for a given nonnegative integer k, there is no regular p-ary bent function f in n variables with w(M_f)=n/2-k ( n+3/2-k, respectively) for an even n ≥ N_p,k (an odd n ≥ N_p,k, respectively), where N_p,k is some positive integer, which is explicitly determined and the w(M_f) of a p-ary function f is some value related to the power of each monomial of f. For the proof of our main results, we use some properties of regular p-ary bent functions, such as the MacWilliams duality, which is proved to hold for regular p-ary bent functions in this paper.