Information Theory

A Hacker's Revenge

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A. Weil. Basic Number Theory, 3rd ed. Springer-Verlag, New York, 1974. L. Welch. Lower bounds on the maximum cross correlation of signals. IEEE Transactions on Information Theory 20(3) (1974), 397–399. N. Y. Yu. New near-optimal codebooks associated with binary Sidel’nikov sequences, in: The Proceedings of IEEE International Symposium on Information Theory Proceedings (ISIT), 2012. N. Y. Yu and G. Gong. Multiplexing realizations of the decimation-Hadamard transform of two-level autocorrelation sequences, in: Coding and Cryptology, volume 5557, pp.

Yu and G. Gong. Multiplexing realizations of the decimation-Hadamard transform of two-level autocorrelation sequences, in: Coding and Cryptology, volume 5557, pp. 248–258. Springer-Verlag, 2009. N. Y. Yu and G. Gong. Multiplicative characters, the Weil bound, and polyphase sequence families with low correlation. IEEE Transactions on Information Theory 56(12) (2010), 6376– 6387. N. Y. Yu and G. Gong. New construction of M -ary sequence families with low correlation from the structure of Sidelnikov sequences.

1, we only need to prove Claim 2: g(x)x M−w = g(x)x w is not a square multiple in Fq [x] where g(x) is defined in Claim 1. However, Claim 1 implies Claim 2. Thus, we only need to show that Claim 1 is true. We now show the case of d = 3. We assume that both f1 and f2 have degree 5. The other cases can be proved in a similar, but much simpler, way. Considering the coefficient of x and constant term, f2 (ατ x) = c · f1 (x) will induce τ = 0 and f1 = f2 . Thus f1 (x) and f2 (ατ x) have at least one different root in F q .

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