SIAM Journal on Algebraic and Discrete Methods
The geometry of quadrics and correlations of sequences
IEEE Transactions on Information Theory
SIAM Journal on Discrete Mathematics
Cyclic codes with few weights and Niho exponents
Journal of Combinatorial Theory Series A
Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity
Power permutations in dimension 32
SETA'10 Proceedings of the 6th international conference on Sequences and their applications
On cyclic codes of length $${2^{2^r}-1}$$ with two zeros whose dual codes have three weights
Designs, Codes and Cryptography
On a conjecture of Helleseth regarding pairs of binary m-sequences
IEEE Transactions on Information Theory
Monomial and quadratic bent functions over the finite fields of odd characteristic
IEEE Transactions on Information Theory
Proof of a conjecture of Sarwate and Pursley regarding pairs of binary m-sequences
IEEE Transactions on Information Theory
A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences
Finite Fields and Their Applications
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Let q be a power of a prime p, let @j"q:F"q-C be the canonical additive character @j"q(x)=exp(2@piTr"F"""q"/"F"""p(x)/p), let d be an integer with gcd(d,q-1)=1, and consider Weil sums of the form W"q","d(a)=@?"x"@?"F"""q@j"q(x^d+ax). We are interested in how many different values W"q","d(a) attains as a runs through F"q^@?. We show that if |{W"q","d(a):a@?F"q^@?}|=3, then all the values in {W"q","d(a):a@?F"q^@?} are rational integers and one of these values is 0. This translates into a result on the cross-correlation of a pair of p-ary maximum length linear recursive sequences of period q-1, where one sequence is the decimation of the other by d: if the cross-correlation is three-valued, then all the values are in Z and one of them is -1. We then use this to prove the binary case of a conjecture of Helleseth, which states that if q is of the form 2^2^^^n, then the cross-correlation cannot be three-valued.