Some Properties of Hadamardtype Matrices on Finite Fields Hadamard matrix is defined as a square matrix where any components are xe2x88x921 or 1, and where any pairs of rows are mutually orthogonal. In this work, we consider the similar matrix on finite field GF(p) where p is an odd prime. In such a matrix, every component is one of the integers on GF(p)0, that is, 1, 2, . . . , pxe2x80x931. Any additions and multiplications should be executed under modulo p. The author has proposed a method to generate such matrices, and applied them to generate nshift orthogonal sequences and complete complementary codes. The generated complete complementary code is a family of multivalued sequences on GF(p)0, where the number of sequence sets, the number of sequences in each sequence set and the sequence length depend on the various divisors of p xe2x80x93 1. Such complete complementary codes with various parameters have not been proposed in previous studies. In this paper, some properties of those matrices are shown to give various construction methods.

Highly nonlinear functions over finite fields Abstract   We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from      F    q    n      to      F    q      to the set of affine functions from      F    q    n      to      F    q     . We prove the conjecture for each q such that the characteristic of      F    q      lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the        q    n    ,  n    1      ReedMuller code over      F    q      and so answers a question raised by Leducq in 2013. Our results extend the case    q    2   , which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory.