A Method to Estimate Partial-Period Correlations
Many applications require large families of sequences with good correlation properties. Some of the best families can be constructed by means of cyclic codes. The full-period correlation of such a family is closely connected with a complete sum of additive characters. In several important special cases it can be easily estimated. On the other hand, the partial period correlations, which are connected with certain incomplete sums of additive characters, are not easy to estimate. A device for estimating is the finite Fourier transform. This approach, which in fact is a modification of an old number theoretic method due to Vinogradov, needs bounds for hybrid sums of additive and multiplicative characters. In this survey we apply this approach in three cases: the m-sequence, the set of dual-BCH sequences, and the small Kasami set.
KeywordsCode Word Cyclic Code Galois Ring Linear Binary Code Number Theoretic Method
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- T. Helleseth and P.V. Kumar, “Sequences with low correlation”, In: Handbook of Coding Theory (eds. V.S. Pless, R.A. Brualdi and W. C. Huffman), to appear.Google Scholar
- I. Honkala and A. Tietäväinen, “Codes and number theory”, In: Handbook of Coding Theory (eds. V.S. Pless, R.A. Brualdi and W. C. Huffman), to appear.Google Scholar
- J. Lahtonen. “On the odd and the aperiodic correlation properties of the Kasami sequences”, IEEE Trans. Information Theory, 41, 1995, 1506 1508.Google Scholar
- R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983.Google Scholar
- A.G. Shanbhag, P.V. Kumar and T. Helleseth, “An upper bound for the aperiodic correlation of weighted-degree CDMA sequences”, Proc. of the 1995 IEEE International Symposium on Information Theory, 1995, 92.Google Scholar
- I.M. Vinogradov, Elements of Number Theory, Dover, 1954.Google Scholar
- A. Weil, “Sur les courbes algébriques et les veriétés qui s’en déduisent”, Actualités Sci. Ind., no. 1041, Hermann, Paris, 1948.Google Scholar