Channel coding is the branch of Information Theory which studies the noise that can occur in data transmitted through a channel. Algebraic Coding Theory is the part of Channel Coding which studies the possibility to detect and correct errors using algebraic and geometric techniques. Nowadays, the best performing linear codes are known to be mostly algebraic geometry codes, also named Goppa codes, which arise from an algebraic curve over a finite field, by the pioneering construction due to V. D. Goppa. The best choices for curves on which Goppa's construction and its variants may provide codes with good parameters are those with many rational points, especially maximal curves attaining the Hasse-Weil upper bound for the number of rational points compared with the genus of the curve. In most cases, the weight distribution of a given code is hard to be computed. Even the problem of computing codewords of minimum weight can be a difficult task, apart from specific cases. In this seminar we investigate the general problem of determining the minimum distance and the weight enumerator polynomial of algebraic-geometric codes. We adopt an approach based on the intersection of the curve defining the code and other curves. We compute the maximal number of intersections that the GK curve can have with a plane curve of degree lower or equal to three. These results are used to determine the minimum distance and the number of minimum weight codewords of dual one-point AG-codes arising from the GK-curve. Afterwards we will focus our attention on the study the possible intersections between the Norm-Trace curve and a parabola. We obtain sharp bounds for the spectrum of the intersections between these two curves. For this purpose, we translate the problem of finding the intersections between these two curves into that of determining the rational points of a certain cubic surfaces. To do so, we employ techniques coming from the properties of irreducible cubic surfaces over finite fields. Then we partially deduce information on the weight enumerator polynomial of the corresponding one-point code.
Referenze Bibliografiche
[1] Abdón, Miriam – Bezerra, Juscelino – Quoos, Luciane, Further examples of maximal curves, Journal of Pure and Applied Algebra 213 (2009) 1192–1196.
[2] Ateş, Leyla – Stichtenoth, Henning, A note on short vectors in lattices from function fields, Finite Fields and Their Applications 39 (2016) 264–271.
[3] Augot, Daniel, Description of minimum weight codewords of cyclic codes by algebraic systems, Finite Fields and their Applications 2 (1996) 138–152.
[4] Ballico, Edoardo – Ravagnani, Alberto, On the duals of geometric goppa codes from norm-trace curves, Finite Fields and Their Applications 20 (2013) 30–39.
[5] Bartoli, Daniele – Bonini, Matteo, Minimum weight codewords in dual algebraic-geometric codes from the giulietti-korchmáros curve, Designs, Codes and Cryptography 87 (2019) 1433–1445.
[6] Bartoli, Daniele – Montanucci, Maria – Zini, Giovanni, Multi point ag codes on the gk maximal curve, Designs, Codes and Cryptography 86 (2018) 161–177.
[7] Bartoli, Daniele – Montanucci, Maria – Zini, Giovanni, Ag codes and ag quantum codes from the ggs curve, Designs, Codes and Cryptography 86 (2018) 2315–2344.
[8] Basili, Bénédicte, Indice de clifford des intersections complètes de l'espace, Bulletin de la Société mathématique de France 124 (1996) 61–95.
[9] Beelen, Peter – Montanucci, Maria, Weierstrass semigroups on the giulietti–korchmáros curve, Finite Fields and Their Applications 52 (2018) 10–29.
[10] Bonini, Matteo – Montanucci, Maria – Zini, Giovanni, On plane curves given by separated polynomials and their automorphisms, Advances in Geometry 20 (2020) 61–70.
[11] Bonini, Matteo – Sala, Massimiliano, Intersections between the norm-trace curve and some low degree curves, arXiv preprint arXiv:1812.08590 (2018).
[12] Browning, TD, The lang–weil estimate for cubic hypersurfaces, Canadian Mathematical Bulletin 56 (2013) 500–502.
[13] Coray, Daniel François – Tsfasman, Michael A, Arithmetic on singular del pezzo surfaces, Proceedings of the London Mathematical Society (1988) 25–87.
[14] Cossidente, A – Korchmáros, G – Torres, F, Curves of large genus covered by the hermitian curve, Communications in Algebra 28 (2000) 4707–4728.
[15] Castellanos, Alonso Sepúlveda – Tizziotti, Guilherme Chaud, Two-point ag codes on the gk maximal curves, IEEE Transactions on Information Theory 62 (2015) 681–686.
[16] Couvreur, Alain, The dual minimum distance of arbitrary-dimensional algebraic–geometric codes, Journal of Algebra 350 (2012) 84–107.
[17] Donati, Giorgio – Durante, Nicola – Korchmáros, Gábor, On the intersection pattern of a unital and an oval in pg (2, q2), Finite Fields and Their Applications 15 (2009) 785–795.
[18] Duursma, Iwan M, Two-point coordinate rings for gk-curves, IEEE transactions on information theory 57 (2011) 593–600.
[19] Fanali, Stefania – Giulietti, Massimo, One-point ag codes on the gk maximal curves, IEEE transactions on information theory 56 (2009) 202–210.
[20] Geil, Olav, On codes from norm–trace curves, Finite fields and their Applications 9 (2003) 351–371.
[21] Farrán, José Ignacio – Munuera, Carlos – Tizziotti, G – Torres, Fernando, Gröbner basis for norm-trace codes, Journal of Symbolic Computation 48 (2013) 54–63.
[22] Fitzgerald, Jeanne – Lax, Robert F, Decoding affine variety codes using gröbner bases, Designs, Codes and Cryptography 13 (1998) 147–158.
[23] Fulton, William, Algebraic curves: an introduction to algebraic geometry, Addison-Wesley (1989).
[24] Garcia, Arnaldo, Curves over finite fields attaining the hasse-weil upper bound, , European Congress of Mathematics (2001).
[25] Garcia, Arnaldo – Güneri, Cem – Stichtenoth, Henning, A generalization of the giulietti–korchmáros maximal curve, Advances in Geometry 10 (2010) 427–434.
[26] Garcia, Arnaldo – Stichtenoth, Henning – Xing, Chao-Ping, On subfields of the hermitian function field, Compositio Mathematica 120 (2000) 137–170.
[27] Giulietti, Massimo – Korchmáros, Gábor, On automorphism groups of certain goppa codes, Designs, Codes and Cryptography 47 (2008) 177–190.
[28] Giulietti, Massimo – Korchmáros, Gábor, A new family of maximal curves over a finite field, Mathematische Annalen 343 (2009) 229.
[29] Goppa, Valerij Denisovič, Geometry and codes, Springer (1988).
[30] Goppa, Valerii Denisovich, Codes on algebraic curves, , Soviet Math. Dokl. 24 (1981).
[31] Goppa, Valerii Denisovich, Algebraico-geometric codes, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 46 (1982) 762–781.
[32] Güneri, Cem – özdemiry, Mehmet – Stichtenoth, Henning, The automorphism group of the generalized giulietti–korchmáros function field, Advances in Geometry 13 (2013) 369–380.
[33] Guralnick, Robert – Malmskog, Beth – Pries, Rachel, The automorphism groups of a family of maximal curves, Journal of Algebra 361 (2012) 92–106.
[34] Hansen, J, Codes on the klein quartic, ideals, and decoding (corresp.), IEEE transactions on information theory 33 (1987) 923–925.
[35] Hartley, Robert William, Determination of the ternary collineation groups whose coefficients lie in the gf (2n), Annals of Mathematics (1925) 140–158.
[36] Hartshorne, Robin, Algebraic geometry, 52 Springer Science & Business Media (2013).
[37] Henn, Hans-Wolfgang, Funktionenkörper mit großer automorphismengruppe., Journal für die reine und angewandte Mathematik 1978 (1978) 96–115.
[38] Hirschfeld, James William Peter – Korchmáros, Gábor – Torres, Fernando – Orihuela, Fernando Eduardo Torres, Algebraic curves over a finite field, Princeton University Press (2008).
[39] Hirschfeld, James William Peter – et al., , Finite projective spaces of three dimensions, Oxford University Press (1985).
[40] Hirschfeld, James, Projective geometries over finite fields, Oxford University Press (1998).
[41] Huppert, Bertram, Endliche gruppen i, 134 Springer-verlag (2013).
[42] Hurt, Norman E, Many rational points: coding theory and algebraic geometry, 564 Springer Science & Business Media (2013).
[43] Ihara, Yasutaka, Some remarks on the number of rational points of algebratic curves over finite fields, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics=
東京大学理学部紀要. 第 1 類 A, 数学 28 (1982) 721–724.
[44] Joyner, David – Ksir, Amy, Automorphism groups of some ag codes, IEEE Transactions on Information Theory 52 (2006) 3325–3329.
[45] Kim, Boran – Lee, Yoonjin, The minimum weights of two-point ag codes on norm-trace curves, Finite Fields and their Applications 53 (2018) 113–139.
[46] Lidl, Rudolf – Niederreiter, Harald, Finite fields, 20 Cambridge university press (1997).
[47] Lang, Serge – Weil, André, Number of points of varieties in finite fields, American Journal of Mathematics 76 (1954) 819–827.
[48] MacWilliams, Florence Jessie – Sloane, Neil James Alexander, The theory of error correcting codes, 16 Elsevier (1977).
[49] Manin, Yu I, Cubic forms: algebra, geometry, arithmetic, Elsevier (1986).
[50] Marcolla, Chiara – Pellegrini, Marco – Sala, Massimiliano, On the hermitian curve and its intersections with some conics, Finite Fields and Their Applications 28 (2014) 166–187.
[51] Marcolla, Chiara – Pellegrini, Marco – Sala, Massimiliano, On the small-weight codewords of some hermitian codes, Journal of Symbolic Computation 73 (2016) 27–45.
[52] Marcolla, Chiara – Roggero, Margherita, Minimum-weight codewords of the hermitian codes are supported on complete intersections, Journal of Pure and Applied Algebra 223 (2019) 3843–3858.
[53] Matthews, Gretchen L, Codes from the suzuki function field, IEEE Transactions on Information Theory 50 (2004) 3298–3302.
[54] Matthews, Gretchen L, Weierstrass semigroups and codes from a quotient of the hermitian curve, Designs, Codes and Cryptography 37 (2005) 473–492.
[55] Miura, S, Geometric goppa codes on some maximal curves and their minimum distance, , Proc. IEEE Workshop on Information Theory (1993).
[56] Munuera, Carlos – Sepúlveda, Alonso – Torres, Fernando, Algebraic geometry codes from castle curves, , Coding Theory and Applications Springer (2008).
[57] Munuera, Carlos – Tizziotti, Guilherme C – Torres, Fernando, Two-point codes on norm-trace curves, , Coding Theory and Applications Springer (2008).
[58] Sala, Massimiliano, Gröbner basis techniques to compute weight distributions of shortened cyclic codes, Journal of Algebra and Its Applications 6 (2007) 403–414.
[59] Segre, Beniamino, Forme e geometrie hermitiane, con particolare riguardo al caso finito, Annali di Matematica Pura ed Applicata 70 (1965) 1–201.
[60] Seidenberg, Abraham, Constructions in algebra, Transactions of the American Mathematical Society 197 (1974) 273–313.
[61] Lachaud, Gilles, Sommes d’eisenstein et nombre de points de certaines courbes algébriques sur les corps finis, CR. Acad. Sci 305 (1987) 729–732.
[62] Stepanov, Serguei A, Codes on algebraic curves, Springer Science & Business Media (2012).
[63] Stichtenoth, Henning, Über die automorphismengruppe eines algebraischen funktionenkörpers von primzahlcharakteristik, Archiv der Mathematik 24 (1973) 527–544.
[64] Stichtenoth, Henning, A note on hermitian codes over gf (q/sup 2/), IEEE transactions on Information Theory 34 (1988) 1345–1348.
[65] Stichtenoth, Henning, Algebraic function fields and codes, 254 Springer Science & Business Media (2009).
[66] Tate, John, Endomorphisms of abelian varieties over finite fields, Inventiones mathematicae 2 (1966) 134–144.
[67] Tiersma, H, Remarks on codes from hermitian curves (corresp.), IEEE transactions on information theory 33 (1987) 605–609.
[68] Tsfasman, Michael A – ă, Serge G – Nogin, Dmitry, Algebraic geometric codes: Basic notions: Basic notions, American Mathematical Soc. (2007).
[69] Walker, Robert J, Algebraic curves, Springer (1978).
[70] Xing, Chaoping – Ling, San, A class of linear codes with good parameters from algebraic curves, IEEE Transactions on Information Theory 46 (2000) 1527–1532.
[71] Xing, Chaoping – Chen, Hao, Improvements on parameters of one-point ag codes from hermitian curves, IEEE Transactions on Information Theory 48 (2002) 535–537.
[72] Yang, Kyeongcheol – Kumar, P Vijay, On the true minimum distance of hermitian codes, , Coding theory and algebraic geometry Springer (1992).
Con il contributo del Ministero per i Beni e le Attività Culturali