An Upper Bound for Weil Exponential Sums over Galois Rings and Applications

IEEE Transactions on Information Theory

Volumn 41, Issue 2, 1995, Pages 456-468

  Kumar, P Vijay ^a   Helleseth, Tor ^b   Calderbank, A R ^c  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b UNIVERSITY OF BERGEN   (Norway)

^c LUCENT TECHNOLOGIES   (United States)

Author keywords

Carlitz Uchiyama bound; Exponential sums; Galois rings; polyphase sequences; sequence design; Weil bound

Indexed keywords

BINARY SEQUENCES; CODES (SYMBOLS); CORRELATION METHODS; CORRELATION THEORY; FUNCTIONS; MATHEMATICAL MODELS; NUMERICAL METHODS; PHASE SHIFT KEYING; POLYNOMIALS;

ALPHABETS; ANALOGS; CARLITZ-UCHIYAMA BOUND; EXPONENTIAL SUMS; FINITE FIELDS; GALOIS RINGS; POLYPHASE SEQUENCES; SEQUENCE DESIGN; WEIL BOUND;

INFORMATION THEORY;

EID: 0029274858     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/18.370147     Document Type: Article

References (27)

1. D. R. Anderson, “A new class of cyclic codes,” SIAM J. Appl. Math., vol. 16, pp. 181–197, 1968.
2. I. F. Blake and J. W. Mark, “A note on complex sequences with low correlations,” IEEE Trans. Inform. Theory, vol. IT-28, no. 5, pp. 814–816, Sept. 1982.
3. S. Boz̧tas, A. R. Hammons, Jr., and P. V. Kumar, “4-phase sequences with near-optimum correlation properties,” IEEE Trans. Inform. Theory, vol. 38, no. 3, pp. 1101–1113, May, 1992.
4. A. R. Calderbank and N. J. A. Sloane, “Modular and p-adic cyclic codes”, submitted to Des., Codes and Cryptogr., 1994.
5. L. Carlitz and S. Uchiyama, “Bounds of exponential sums,” Duke Math. J., pp. 37–41, 1957.
6. C. Chevalley, Introduction to the Theory of Algebraic Functions of One Variable. New York: AMS Math. Surveys, 1951.
7. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z4-linearity of Kerdock, Preparata, Goethals and related codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 301–319, Mar. 1994.
8. T. Helleseth, “On the covering radius of cyclic linear codes and arithmetic codes,” Discr. Appl. Math., vol. 11, pp. 157–173, 1985.
9. N. Koblitz, “p-adic numbers, p-adic analysis, and zeta functions,” in Graduate Texts in Mathematics, vol. 58. New York: Springer-Verlag, 1977.
10. P. V. Kumar, T. Helleseth, A. R. Calderbank, and A. R. Hammons, Jr., “Large sequence families with good correlation for CDMA,” submitted to IEEE Trans. Inform. Theory, also in Proc. 1994 Int. Symp. un Information Theory (Trondheim, Norway, June 27–July 2, 1994).
11. P. V. Kumar and O. Moreno, “Polyphase sequences with periodic correlation properties better than binary sequences,” IEEE Trans. Inform. Theory, vol. 37, no. 3, pp. 603–616, May, 1991.
12. R. Lidl and H. Niederreiter, Finite-Fields, vol. 20 of Encyclopedia of Mathematics and Its Applications. Reading, MA: Addison-Wesley, 1983.
13. R. A. Liebler and R. A. Mena. “Certain distance-regular digraphs and related rings of characteristic 4,” J. Combinatorial Theory, vol. 47, pp. 111–123, 1988.
14. B. R. MacDonald, Finite Rings with Identity. New York: Marcel Dekker, 1974.
15. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
16. C. J. Moreno, “Algebraic curves over finite fields,” in Tracts in Mathematics, vol. 97. Cambridge, England: Cambridge Univ. Press, 1991.
17. A. A. Nechaev, “The Kerdock code in a cyclic form,” Discr. Math., vol. 1, pp. 123–139, 1989.
18. D. V. Sarwate and M. B. Pursley, “Cross-correlation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, no. 5, pp. 593–618, May 1980.
19. P. Shankar, “On BCH codes over arbitrary integer rings,” IEEE Trans. Inform. Theory, vol. IT-25, no. 3, pp. 480–483, May, 1979.
20. V. M. Sidelnikov, “On mutual correlation of sequences,” Soviet Math.-Dokl., vol. 12, no. 1, pp. 197–201, 1971.
21. M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread-Spectrum Communications Handbook, rev. ed. New York: McGraw-Hill, 1994.
22. P. Solé, “A quaternary cyclic code and a family of quadriphase sequences with low correlation properties,” in Lecture Notes in Computer Science, vol. 388, 1989, pp. 193–201.
23. H. Stichtenoth, Algebraic Function Fields and Codes. New York: Springer-Verlag, Universitext, 1993.
24. A. Tietäväinen, “On the covering radius of long BCH codes,” Discr. Appl. Math., vol. 16, pp. 75–77, 1987.
25. P. Udaya and M. U. Siddiqi, “Large linear complexity sequences over Z4 for quadriphase modulated communication systems having good correlation properties,” in Proc. IEEE Int. Symp. on Information Theory (Budapest, Hungary, June 24–28, 1991).
26. J. V. Uspensky, Theory of Equations. New York: McGraw-Hill, 1948.
27. A. Weil, “On some exponential sums,” Proc. Nat. Acad. Sci., vol. 34, pp. 204–207, 1948.