Weight distributions of geometric Goppa codes

Duursma, Iwan

doi:10.1090/S0002-9947-99-02179-0

Weight distributions of geometric Goppa codes
HTML articles powered by AMS MathViewer

by Iwan M. Duursma PDF

Trans. Amer. Math. Soc. 351 (1999), 3609-3639 Request permission

Abstract:

The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results restrict to codes from elliptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of $L$-series. The results include general properties of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three.

References

E. Artin and J.T. Tate. Class field theory. Math. Lecture Notes. Benjamin, New York, 1967.
E. Artin and J. Tate, Class field theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0223335
Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
Richard E. Blahut, Theory and practice of error control codes, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. MR 698946
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989. MR 1002568, DOI 10.1007/978-3-642-74341-2
Yves Driencourt and Jean-Francis Michon, Elliptic codes over fields of characteristic $2$, J. Pure Appl. Algebra 45 (1987), no. 1, 15–39. MR 884628, DOI 10.1016/0022-4049(87)90081-8
I.M. Duursma. Average weight enumerators for geometric goppa codes. in: ACCT-Proceedings, p.82. Novgorod, Russia, 1994.
G. David Forney Jr., Concatenated codes, MIT Research Monograph, No. 37, The M.I.T. Press, Cambridge, Mass., 1966. MR 0347434
Gerhard Frey and Hans-Georg Rück, A remark concerning $m$-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comp. 62 (1994), no. 206, 865–874. MR 1218343, DOI 10.1090/S0025-5718-1994-1218343-6
A. Fröhlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR 1215934
V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259 (1981), no. 6, 1289–1290 (Russian). MR 628795
H. Heilbronn, Zeta-functions and $L$-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 204–230. MR 0218327
The theory of numbers (1975), xi+541. With contributions by T. Tannaka, T. Tamagawa, I. Satake, Akira Hattori, G. Fujisaki and H. Shimizu; Translated from the Japanese by K. Iyanaga; North-Holland Mathematical Library, Vol. 8. MR 0444603
T. Kasami, S. Lin and W.W. Peterson. Some Results on Weight Distributions of BCH Codes. IEEE Trans. Inf. Theor., 12:274, 1966.
G. L. Katsman and M. A. Tsfasman, Spectra of algebro-geometric codes, Problemy Peredachi Informatsii 23 (1987), no. 4, 19–34 (Russian). MR 931714
Torleiv Kløve, The weight distribution of linear codes over $\textrm {GF}(q^{l})$ having generator matrix over $\textrm {GF}(q)$, Discrete Math. 23 (1978), no. 2, 159–168. MR 523410, DOI 10.1016/0012-365X(78)90114-0
Wen-Ch’ing Winnie Li, Character sums and abelian Ramanujan graphs, J. Number Theory 41 (1992), no. 2, 199–217. With an appendix by Ke Qin Feng and the author. MR 1164798, DOI 10.1016/0022-314X(92)90120-E
F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. I, North-Holland Mathematical Library, Vol. 16, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0465509
A.J. Menezes, I.F. Blake, R.C. Mullin, X. Gao, S.A. Vanstone, and T. Yaghoobian. Applications of finite fields, volume 0199 of SECS. Kluwer Acad. Publ., Dordrecht, The Netherlands, 1993.
Carlos Moreno, Algebraic curves over finite fields, Cambridge Tracts in Mathematics, vol. 97, Cambridge University Press, Cambridge, 1991. MR 1101140, DOI 10.1017/CBO9780511608766
Ruud Pellikaan, On the gonality of curves, abundant codes and decoding, Coding theory and algebraic geometry (Luminy, 1991) Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992, pp. 132–144. MR 1186421, DOI 10.1007/BFb0087998
Ruud Pellikaan, On special divisors and the two variable zeta function of algebraic curves over finite fields, Arithmetic, geometry and coding theory (Luminy, 1993) de Gruyter, Berlin, 1996, pp. 175–184. MR 1394933
Michael Rosen, The Hilbert class field in function fields, Exposition. Math. 5 (1987), no. 4, 365–378. MR 917350
Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961
Michael A. Tsfasman, Global fields, codes and sphere packings, Astérisque 198-200 (1991), 373–396 (1992). Journées Arithmétiques, 1989 (Luminy, 1989). MR 1144339
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric codes, Mathematics and its Applications (Soviet Series), vol. 58, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian by the authors. MR 1186841, DOI 10.1007/978-94-011-3810-9
Michael A. Tsfasman and Serge G. Vlăduţ, Geometric approach to higher weights, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1564–1588. Special issue on algebraic geometry codes. MR 1391017, DOI 10.1109/18.476213
Jacobus Hendricus van Lint, Introduction to coding theory, Problemy Matematicheskogo Analiza [Problems in Mathematical Analysis], vol. 8, Springer-Verlag, New York-Berlin, 1982. MR 658134, DOI 10.1007/978-3-662-07998-0
Jacobus H. van Lint and Gerard van der Geer, Introduction to coding theory and algebraic geometry, DMV Seminar, vol. 12, Birkhäuser Verlag, Basel, 1988. MR 1015293, DOI 10.1007/978-3-0348-9286-5
S. G. Vlăduţ, Two remarks on the spectra of algebraic geometry codes, Arithmetic, geometry and coding theory (Luminy, 1993) de Gruyter, Berlin, 1996, pp. 253–261. MR 1394938
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
J. Weissinger. Theorie der Divisorenkongruenzen. Abh. Math. Sem. Hanischen Univ., 12:115-126, 1937.
Victor K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37 (1991), no. 5, 1412–1418. MR 1136673, DOI 10.1109/18.133259