Binary m-sequences with three-valued crosscorrelation: A proof of Welch's conjecture
نویسندگان
چکیده
We prove the long-standing conjecture of Welch stating that for odd = 2 + 1, the power function with = 2 + 3 is maximally nonlinear on GF (2 ) or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequences of degree and a decimation of that sequence by 2 + 3 takes on precisely the three values 1 1 2 .
منابع مشابه
ar X iv : c s / 06 08 12 3 v 1 [ cs . I T ] 3 1 A ug 2 00 6 Proof of a Conjecture of Helleseth Regarding Pairs of Binary m - Sequences ∗
–Binary m-sequences are maximal length sequences generated by shift registers of length m, that are employed in navigation, radar, and spread-spectrum communication. It is well known that given a pair of distinct m-sequences, the crosscorrelation function must take on at least three values. This correspondence shows the three correlation values are symmetric about -1. The main result is a proof...
متن کاملF eb 2 00 7 Characterization of m - Sequences of Lengths 2 2 k − 1 and 2 k − 1 with Three - Valued Crosscorrelation
Considered is the distribution of the crosscorrelation between m-sequences of length 2−1, where m = 2k, and m-sequences of shorter length 2 − 1. New pairs of m-sequences with three-valued crosscorrelation are found and the complete correlation distribution is determined. Finally, we conjecture that there are no more cases with a three-valued crosscorrelation apart from the ones proven here.
متن کاملCharacterization of m-Sequences of Lengths $2^{2k}-1$ and $2^k-1$ with Three-Valued Crosscorrelation
Considered is the distribution of the crosscorrelation between m-sequences of length 2−1, where m = 2k, and m-sequences of shorter length 2 − 1. New pairs of m-sequences with three-valued crosscorrelation are found and the complete correlation distribution is determined. Finally, we conjecture that there are no more cases with a three-valued crosscorrelation apart from the ones proven here.
متن کاملProof of a Conjecture of Helleseth: Maximal Linear Recursive Sequences of Period 22n-1 Never Have Three-Valued Cross-Correlation
We prove a conjecture of Helleseth that claims that for any n ≥ 0, a pair of binary maximal linear sequences of period 22 n −1 can not have a three-valued cross-correlation function.
متن کاملWelch's Bound and Sequence Sets for Code-Division Multiple-Access Systems
Welch's bound for a set of M complex equi-energy sequences is considered as a lower bound on the sum of the squares of the magnitudes of the inner products between all pairs of these sequences. It is shown that, when the sequences are binary (±-1 valued) sequences assigned to the M users in a synchronous code-division multiple-access (S-CDMA) system, precisely such a sum determines the sum of t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- IEEE Trans. Information Theory
دوره 46 شماره
صفحات -
تاریخ انتشار 2000