A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences

Finite Fields and their Applications

Volumn 7, Issue 2, 2001, Pages 253-286

  Hollmann, Henk D L ^a   Xiang, Qing ^b  

^a PHILIPS RESEARCH LABORATORIES   (Netherlands)

^b UNIVERSITY OF DELAWARE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038410824     PISSN: 10715797     EISSN: None     Source Type: Journal    
DOI: 10.1006/ffta.2000.0281     Document Type: Article

References (22)

1. Brouwer A. E., Tolhuizen L. M. G. M. A sharpening of the Johnson bound for binary linear codes. Des. Codes and Cryptogr. 3:1991;95-98.
2. Calderbank A. R., Goethals J.-M. Three-weight codes and association schemes. Philips J. Res. 39:1984;143-152.
3. Calderbank A. R., McGuire G., Poonen B., Rubinstein M. On a conjecture of Helleseth regarding pairs of binary m-sequences. IEEE Trans. Inform. Theory. 42:1996;988-990.
4. A. Canteaut, P. Charpin, and, H. Dobbertin, Binary m-sequences with three-valued cross-correlation: A proof of Welch's conjecture, submitted for publication.
5. Carlet A., Charpin P., Zinoviev V. Codes, bent functions, and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15:1998;125-156.
6. Chang A., Gaal P., Golomb S. W., Gong G., Kumar P. V. On a Sequence Conjectured to Have Ideal 2-level Autocorrelation Function. 1998;ISIT, Cambridge.
7. Cusick T., Dobbertin H. Some new three-valued cross-correlation functions for binary m-sequences. IEEE Trans. Inform. Theory. 42:1996;1238-1240.
8. n): The Welch case. IEEE Trans. Inform. Theory. 45:1999;1271-1275.
9. n): The Niho case. Inform. and Comput. 151:1999;57-72.
10. Evans R., Hollmann H. D. L., Krattenthaler C., Xiang Q. Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets. J. Combin. Theory Ser. A. 87:1999;74-119.
11. Gold R. Maximal recursive sequences with 3-valued cross-correlation functions. IEEE Trans. Inform. Theory. 14:1968;154-156.
12. Helleseth T. Some results about the cross-correlation function between two maximal linear sequences. Discrete Math. 16:1976;209-232.
13. Kasami T. Weight distribution of Bose-Chaudhuri-Hocquenghem codes. Bose R. C., Dowling T. A. Proceedings Conference Combinatorial Mathematics and Its Applications. 1969;335-357 Univ. of North Carolina Press, Chapel Hill.
14. Kasami T. The weight enumerator for several classes of subcodes of the 2nd order binary Reed-Muller codes. Inform. and Control. 18:1971;369-394.
15. van Lint J. H., Wilson R. M. On the minimum distance of cyclic codes. IEEE Trans. Inform. Theory. 32:1986;23-40.
16. McEliece R. J. On periodic sequences from GF(q). J. Combin. Theory Ser. A. 10:1971;80-91.
17. McGuire G., Calderbank A. R. Proof of a conjecture of Sarwate and Pursley regarding pairs of binary m-sequences. IEEE Trans. Inform. Theory. 41:1995;1153-1155.
18. MacWilliams F. J., Sloane N. J. A. The Theory of Error-Correcting Codes. 1977;North-Holland, Amsterdam.
19. Niho Y. Multi-valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences. 1972;Univ. of Southern California, Los Angeles.
20. Pless V. Power moment identities on weight distributions in error correcting codes. Inform. and Control. 6:1963;147-152.
21. Sarwate D. V., Pursley M. B. Cross-correlation properties of pseudorandom and related sequences. Proc. IEEE. 68:1980;593-619.
22. Welch L. R. Trace Mappings in Finite Fields and Shift Register Cross-Correlation Properties. 1969;Univ. Southern CaliforniaElectrical Engineering Department Report.