A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences

Binary m-sequences are widely applied in navigation, radar, and communication systems because of their nice autocorrelation and cross-correlation properties. In this paper, we consider the cross-correlation between a binary m-sequence of length 2m−1 and a decimation of that sequence by an integer t. We will be interested in the number of values attained by such cross-correlations. As is well known, this number equals the number of nonzero weights in the dual of the binary cyclic code C1,t of length 2m−1 with defining zeros α and αt, where α is a primitive element in GF(2m). There are many pairs (m, t) for which C 1,t is known or conjectured to have only few nonzero weights. The three-weight examples include the following cases: We present a method of proving many of these known or conjectured results, including all of the above cases, in a unified way.