Bent functions are Boolean functions in an even number of variables that are indicators of Hadamard difference sets in elementary abelian 2-groups. A bent functionfinmvariables is said to be normal if it is constant on an affine space of dimensionm/2. In this paper, we demonstrate that all bent functions in m = 8 variables, whose exact count, determined by Langevin and Leander (Des. Codes Cryptogr. 59(1–3): 193–205, 2011), is approximately 2106, share a common algebraic property: every 8-variable bent function is normal, up to the addition of a linear function. With this result, we complete the analysis of the normality of bent functions for the last unresolved case,m= 8. It is already known that all bent functions inmvariables are normal form≤ 6, while form≥ 10, there exist bent functions that cannot be made normal by adding linear functions. Consequently, we provide a complete solution to an open problem by Charpin (J. Complex. 20(2-3): 245-265, 2004).

In this paper we report the following computational results on partial spread functions in eight variables: (i) the numbers of equivalence classes of partial spread functions (in eight variables) of all possible orders; (ii) the total number of partial spread bent functions in eight variables; (iii) the distribution of the cardinalities of stabilizers (in GL(8, F2)) of partial spread bent functions in eight variables. The computational method is also described.

A conjecture of Niho states that under certain assumptions the Fourier transform of the function${\rm Tr}(x^{d})$on$\BBF _{2^{n}}$, where$d=(2^{tk}+1)/(2^{k}+1)$, has a spectrum with at most five values. We present a counterexample to this conjecture, and the theory behind finding it. We use the theory of quadratic forms over$\BBF _{2}$.

We describe a new invariant that we have used to obtain the complete classification of the cubic forms of nine variables. In particular, we compute the covering radius of the Reed-Muller code RM(2, 9) into RM(3, 9).