Abstract
New families of unimodular sequences of length p = 3f + 1 with zero autocorrelation are described, p being a prime. The construction is based on employing Gauss periods. It is shown that in this case elements of the sequences are algebraic numbers defined by irreducible polynomials over ℤ of degree 12 (for the first family) and 6 (for the second family). In turn, these polynomials are factorized in some extension of the field ℚ into polynomials of degree, respectively, 4 and 2, which are written explicitly. For p = 13, using the exhaustive search method, full classification of unimodular sequences with zero autocorrelation is given.
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REFERENCES
Gabidulin, E.M., On Classification of Sequences with the Perfect Periodic Auto-Correlation Function, in Proc. 3rd Int. Colloq. on Coding Theory, Dilijan, Armenia, 1990, Yerevan, 1991, pp. 24-30.
Gabidulin, E.M., There Are Only Finitely Many Perfect Auto-Correlation Polyphase Sequences of Prime Length, in Proc. 1994 IEEE Int. Sympos. on Information Theory, Trondheim, Norway, 1994, pp. 282.
van der Waerden, B.L., Algebra, Berlin: Springer, 1971, 8 ed. Translated under the title Algebra, Moscow: Nauka, 1976.
Prasolov, V.V., Mnogochleny (Polynomials), Moscow: Mos. Tsentr Nepreryvnogo Mat. Obrazovaniya, 2001.
Cox, D.A., Little, J., and O'Shea, D., Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer, 1997, 2nd ed. Translated under the title Idealy, mnogoobraziya i algoritmy: Vvedenie v vychislitel'nye aspekty algebraicheskoi geometrii i kommutativnoi algebry, Moscow: Mir, 2000.
Edwards, H.M., Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, New York: Springer, 1977. Translated under the title Poslednyaya teorema Ferma, Moscow: Mir, 1980.
Fan, P. and Darnell, M., Sequences Design for Communications Applications, Taunton: RSP; New York: Wiley, 1996.
Frank, R.L., Polyphase Codes with Good Nonperiodic Correlation Properties, IEEE Trans. Inf. Theory, 1963, vol. 9, no. 1, pp. 43-45.
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Gabidulin, E.M., Shorin, V.V. New Sequences with Zero Autocorrelation. Problems of Information Transmission 38, 255–267 (2002). https://doi.org/10.1023/A:1022093728009
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DOI: https://doi.org/10.1023/A:1022093728009