In this paper, we consider Cusick's method to find the number of values of the polynomials $f_a(x)=x^{a}\,(x+1)^{2^k-1}$, when $x\in GF(2^{2k})$. We will prove that under certain conditions $f_a(x)$ and $f_{2-a}(x)$ have the same number of values. We will also prove a conjecture due to Cusick.

Suppose that n=2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d=(2/sup k/-1)s+1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s=3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations.

Consider the cross-correlation function Cd(tau) between two m-sequences of period p2k-1 that differ by a decimation d. Assume that d is of Niho type and Cd(tau) is four-valued. In this correspondence, the values of Cd(tau) are described. If p=2 then the values of Cd(tau) are -1-2k,-1,-1+2k, and -1+2k+jfor some integer j>0 for which 2j divides k. If p>2 then the values are -1-pk,-1,-1+pk, and -1+2middotpk

A covering code construction is presented. Using this construction it is shown that t[52,39]=3, t[36,21]=4, t[58,32]=7, K(32,2)/spl les/62/spl middot/2/sup 18/, and K(62,5)/spl les/31/spl middot/2/sup 37/, where t[n,k] is the minimum covering radius among all binary [n,k] codes and K(n,R) is the minimum cardinality of a binary code of length n and covering radius R. Four new linear codes found by computer search are also given. These include a [23,9]5 code, a [32,8]10 code, a [51,41]2 code, and a [45,20]8 code.