A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences
✍ Scribed by Henk D.L. Hollmann; Qing Xiang
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 412 KB
- Volume
- 7
- Category
- Article
- ISSN
- 1071-5797
No coin nor oath required. For personal study only.
✦ Synopsis
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 2K!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 C R of length 2K!1 with de"ning zeros and R, where is a primitive element in GF(2K). There are many pairs (m, t) for which C, R is known or conjectured to have only few nonzero weights. The three-weight examples include the following cases:
(a) t"1#2P, if m/(m, r) odd, (b) t"2P!2P#1, if m/(m, r) odd, (c) m"2r#1 odd, t"2P#3, and (d) m odd, 4r,!1 mod m, t"2P#2P!1. We present a method of proving many of these known or conjectured results, including all of the above cases, in a uni"ed way.