In this paper, we investigate the properties of generalized bent functions defined on Z n 2 with values in Z q , where q ≥ 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana-McFarland type bent functions and Dillon's functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana-McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on Z n 2 with values in Z 4 and Z 8 . Moreover, we propose several constructions of such generalized bent functions for both n even and n odd.
Under mild conditions on n, p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF (p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that X(2 t , 2 t+1 l − 1) are the only nonlinear balanced elementary symmetric polynomials over GF (2), whereand we prove various results in support of this conjecture.
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