LetKqn(a) be a Kloosterman sum over the finite fieldFqnof characteristicp. In this note so called subfield conjecture is proved: ifane 0 belongs to the proper subfieldFqofFqn, thenKqn(a) ne -1. This completes recent works on the subfield conjecture by Shparlinski, and Moisio and Lisonek. The problem is motivated by some applications to bent functions. Moreover, in the course of the proof a large class of translates of Dickson polynomials are shown to be irreducible.

Binary hyper-Kloosterman codes $C(r,m)$ of length $(2^r-1)^{m-1}$ are a quasi-cyclic generalization of the dual of the Melas code of length $2^r-1$. In this paper the duals $C^{\perp}(r,m)$, i.e., a generalization of the Melas code $C^{\perp}(r,2)$ itself, are studied. In particular, the minimum distance of $C^{\perp}(r,m)$ for all $r,m\ge2$, the weight distribution of $C(2,m)$ and $C^{\perp}(2,m)$ for all $m\ge2$, and the weight distribution of $C(r,3)$ and $C^{\perp}(r,3)$ for all $r\ge2$ are obtained.

Frequency-hopping (FH) spread spectrum and direct-sequence spread spectrum are two main spread-coding technologies. Frequency-hopping sequences are needed in FH code-division multiple-access (CDMA) systems. In this correspondence, three classes of optimal frequency-hopping sequences are constructed with algebraic methods. The three classes are based on perfect nonlinear functions, power functions, and norm functions, respectively. Both individual optimal frequency-hopping sequences and optimal families of frequency-hopping sequences are presented.

In this correspondence, we give the moments of a Kloosterman sum over Fqin terms of the frequencies of weights in the binary Zetterberg code of length q+1, which are known by the work of Schoof and van der Vlugt. The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As a corollary the weight distribution of a Zetterberg-type binary cyclic code is obtained

Two recursive algorithms for computing the weight distributions of certain binary irreducible cyclic codes of length n in the so-called index 2 case are presented. The running times of these algorithms are smaller than O(log/sup 2/r) where r=2/sup m/ and n is a factor of r-1.