Abstract
We consider the rank polynomial of a matroid and some well-known applications to graphs and linear codes. We compare rank polynomials with two-variable zeta functions for algebraic curves. This leads us to normalize the rank polynomial and to extend it to a rational rank function. As applications to linear codes we mention: A formulation of Greeneās theorem similar to an identity for zeta functions of curves first found by Deninger, the definition of a class of generating functions for support weight enumerators, and a relation for algebraic-geometric codes between the matroid of a code and the two-variable zeta function of a curve.
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References
Ashikhmin, A., Barg, A.: Binomial moments of the distance distribution: bounds and applications. IEEE Trans. Inform. TheoryĀ 45(2), 438ā452 (1999)
Baldassarri, F., Deninger, C., Naumann, N.: A motivic version of Pellikaanās two variable zeta function. arXiv:math.AG/0302121
Barg, A.: The matroid of supports of a linear code. Appl. Algebra Engrg. Comm. Comput.Ā 8(2), 165ā172 (1997)
Barg, A.: On some polynomials related to weight enumerators of linear codes. SIAM J. Discrete Math.Ā 15(2), 155ā164 (2002) (electronic)
Bass, H.: The Ihara-Selberg zeta function of a tree lattice. Internat. J. Math.Ā 3(6), 717ā797 (1992)
BollobĆ”s, B.: Modern graph theory. Graduate Texts in Mathematics, vol.Ā 184. Springer, New York (1998)
Brylawski, T., Oxley, J.: The Tutte polynomial and its applications. In: Matroid applications. Encyclopedia Math. Appl., vol.Ā 40, pp. 123ā225. Cambridge Univ. Press, Cambridge (1992)
Chung, F.R.K.: Spectral graph theory. CBMS Regional Conference Series in Mathematics, vol.Ā 92. Published for the Conference Board of the Mathematical Sciences, Washington (1997)
Crapo, H.H., Rota, G.-C.: On the foundations of combinatorial theory: Combinatorial geometries. The MIT Press, Cambridge (1970) (preliminary edition)
Deninger, C.: Two-variable zeta functions and regularized products. arXiv:math.NT/0210269
Duursma, I.: From weight enumerators to zeta functions. Discrete Appl. Math.Ā 111(1-2), 55ā73 (2001)
Duursma, I.M.: Results on zeta functions for codes. In: Proceedings: Fifth Conference on Algebraic Geometry, Number Theory, Coding Theory and Cryptography, University of Tokyo, January 17-19 (2003); arXiv:math.CO/0302172
Duursma, I.M.: Zeta functions for linear codes (1997) (preprint)
Duursma, I.M.: Weight distributions of geometric Goppa codes. Trans. Amer. Math. Soc.Ā 351(9), 3609ā3639 (1999)
Edmonds, J.: Lehmanās switching game and a theorem of Tutte and Nash- Williams. J. Res. Nat. Bur. Standards Sect. BĀ 69B, 73ā77 (1965)
Godsil, C., Royle, G.: Algebraic graph theory. Graduate Texts in Mathematics, vol.Ā 207. Springer, New York (2001)
Greene, C.: Weight enumeration and the geometry of linear codes. Studies in Appl. Math.Ā 55(2), 119ā128 (1976)
Helleseth, T., KlĆøve, T., Mykkeltveit, J.: The weight distribution of irreducible cyclic codes with block length n 1((q lā1)/N). Discrete Math.Ā 18(2), 179ā211 (1977)
Hoffman, J.W.: Remarks on the zeta function of a graph. LSU Mathematics Electronic Preprint Series 2002-17 (2002)
Jaeger, F.: On Tutte polynomials of matroids representable over GF(q). European J. Combin.Ā 10(3), 247ā255 (1989)
KlĆøve, T.: The weight distribution of linear codes over GF(ql) having generator matrix over GF(q). Discrete Math.Ā 23(2), 159ā168 (1978)
KlĆøve, T.: Support weight distribution of linear codes. Discrete Math.Ā 106(107), 311ā316 (1992); A collection of contributions in honour of Jack van Lint
MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. North-Holland Mathematical Library, vol.Ā 16. North-Holland Publishing Co., Amsterdam (1977)
Moreno, C.: Algebraic curves over finite fields. Cambridge Tracts in Mathematics, vol.Ā 97. Cambridge University Press, Cambridge (1991)
Munuera, C.: On the generalized Hamming weights of geometric Goppa codes. IEEE Trans. Inform. TheoryĀ 40(6), 2092ā2099 (1994)
Naumann, N.: On the irreducibility of the two variable zeta-function for curves over finite fields. arXiv:math.AG/0209092
Oxley, J.: What is a matroid? LSU Mathematics Electronic Preprint Series 2002-9 (2002)
Oxley, J.G.: Matroid theory. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1992)
Pellikaan, R.: On the gonality of curves, abundant codes and decoding. In: Coding theory and algebraic geometry, Luminy, 1991. Lecture Notes in Math., vol.Ā 1518, pp. 132ā144. Springer, Berlin (1992)
Pellikaan, R.: On special divisors and the two variable zeta function of algebraic curves over finite fields. In: Arithmetic, geometry and coding theory, Luminy, 1993, pp. 175ā184. de Gruyter, Berlin (1996)
Rosenstiehl, P., Read, R.C.: On the principal edge tripartition of a graph. Ann. Discrete Math.Ā 3, 195ā226 (1978); Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977)
Simonis, J.: The effective length of subcodes. Appl. Algebra Engrg. Comm. Comput.Ā 5(6), 371ā377 (1994)
Stichtenoth, H.: Algebraic function fields and codes. Universitext. Springer, Berlin (1993)
Tsfasman, M.A., VlÄduÅ£, S.G.: Geometric approach to higher weights. IEEE Trans. Inform. TheoryĀ 41(6, part 1), 1564ā1588 (1995); Special issue on algebraic geometry codes
Tutte, W.T.: A ring in graph theory. Proc. Cambridge Philos. Soc.Ā 43, 26ā40 (1947)
Tutte, W.T.: On the algebraic theory of graph colorings. J. Combinatorial TheoryĀ 1, 15ā50 (1966)
van der Geer, G., Schoof, R.: Effectivity of Arakelov divisors and the theta divisor of a number field. Selecta Math. (N.S.)Ā 6(4), 377ā398 (2000)
van Lint, J.H.: Introduction to coding theory, 3rd edn. Graduate Texts in Mathematics, vol.Ā 86. Springer, Berlin (1999)
Wei, V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inform. TheoryĀ 37(5), 1412ā1418 (1991)
Weil, A.: Sur les courbes algĆ©briques et les variĆ©tĆ©s qui sāen dĆ©duisent. ActualitĆ©s Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. StrasbourgĀ 7 (1945); Hermann et Cie., Paris (1948)
Welsh, D.J.A.: Matroid theory. L. M. S. Monographs, vol.Ā 8. Academic Press [Harcourt Brace Jovanovich Publishers], London (1976)
Yang, K., Vijay Kumar, P., Stichtenoth, H.: On the weight hierarchy of geometric Goppa codes. IEEE Trans. Inform. TheoryĀ 40(3), 913ā920 (1994)
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Duursma, I.M. (2004). Combinatorics of the Two-Variable Zeta Function. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds) Finite Fields and Applications. Fq 2003. Lecture Notes in Computer Science, vol 2948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24633-6_9
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DOI: https://doi.org/10.1007/978-3-540-24633-6_9
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