Differential approach for the study of duals of algebraic-geometric codes on surfaces

Couvreur, Alain

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 95-120.

The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code $C_L(D,G)$ on a curve is the differential code $C_{\Omega }(D,G)$ . We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples of codes on surfaces, and in particular surfaces with Picard number $1$ like elliptic quadrics or some particular cubic surfaces. The parameters of some of the studied codes reach those of the best known codes up to now.

L’objet de cet article est l’étude des orthogonaux de codes fonctionnels sur des surfaces algébriques. Nous en donnons une description géométrique directe à l’aide de formes différentielles. Bien que moins élémentaire, cette approche peut être vue comme une extension naturelle aux surfaces du résultat affirmant que l’orthogonal d’un code $C_L(D,G)$ sur une courbe est le code différentiel $C_{\Omega }(D,G)$. Nous étudions les paramètres de ces codes et établissons un résultat de minoration de leur distance minimale. À l’aide de cette borne, on peut étudier certains exemples de codes sur des surfaces, en particulier sur des surfaces de nombre de Picard égal à $1$ comme les quadriques elliptiques ou certaines surfaces cubiques. Les paramètres de certains codes étudiés égalent ceux des meilleurs codes connus à l’heure actuelle.

[1] Y. Aubry, Reed-Muller codes associated to projective algebraic varieties. Lecture Notes in Math. 1518 (1992), 4–17. | MR | Zbl

[2] Y. Aubry, M. Perret, On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields. Finite Fields Appl. 10(3) (2004), 412–431. | MR | Zbl

[3] A. Couvreur, Sums of residues on algebraic surfaces and application to coding theory. J. of Pure and Appl. Algebra 213 (2009), 2201–2223. | MR | Zbl

[4] A. Couvreur, Résidus de $2$-formes différentielles sur les surfaces algébriques et applications aux codes correcteurs d’erreurs. PhD thesis, Inst. Math. Toulouse (2008). ArXiv:0905.2341.

[5] P. Delsarte, J.-M. Goethals, F. J. MacWilliams, On generalized Reed-Muller codes and their relatives. Information and Control. 16 (1970), 403–442. | MR | Zbl

[6] I. Duursma, C.Y. Chen, Geometric Reed-Solomon codes of length 64 and 65 over ${\mathbf{F}}_8$. IEEE Trans. Inform. Theory 49(5) (2003), 1351–1353. | MR | Zbl

[7] F. A. B. Edoukou, Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen’s conjecture. Finite Fields Appl. 13(3) (2007), 616–627. | MR | Zbl

[8] F. A. B. Edoukou, Codes defined by forms of degree 2 on quadric surfaces. IEEE Trans. Inform. Theory 54(2) (2008), 860–864. | MR

[9] V. D. Goppa, Codes on algebraic curves. Dokl. Akad. Nauk SSSR. 259(6) (1981), 1289–1290. | MR | Zbl

[10] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, (2007). Accessed on 2010-11-15, http://www.codetables.de.

[11] R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics, 1977. | MR | Zbl

[12] J. Kollàr, K. E. Smith, A. Corti, Rational and nearly rational varieties. Cambridge University Press, 2004. | MR | Zbl

[13] F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. North-Holland Mathematical Library, 1977. | Zbl

[14] Martinínez-Moro, Edgar and Munuera, Carlos and Ruano, Diego, Advances in Algebraic Geometry Codes. World Scientific, 2008. | MR

[15] A. N. Paršin, On the arithmetic of two-dimensional schemes. I. Distributions and residues. Izv. Akad. Nauk SSSR Ser. Mat. 40(4) (1976), 736–773. | MR | Zbl

[16] V. S. Pless, W. C. Huffman, R. A. Brualdi, Handbook of coding theory. North-Holland Mathematical Library, 1998. | MR

[17] B. Poonen, Bertini theorems over finite fields. Ann. of Math. 160(3) (2004), 1099–1127. | MR | Zbl

[18] R. Schürer, W. C. Schmid, MinT: a database for optimal net parameters. In Monte Carlo and Quasi-Monte Carlo Methods, (2006), 457–469. Available online on http://mint.sbg.ac.at. | MR | Zbl

[19] J.-P. Serre, Lettre à M. Tsfasman. Astérisque (198-200) (1992), 351–353. | Numdam | MR | Zbl

[20] H. Stichtenoth, Algebraic function fields and codes. Universitext. Springer-Verlag, 1993. | MR | Zbl

[21] H. P. F. Swinnerton-Dyer, The Zeta function of a cubic surface over a finite field. Proc. Cambridge Philos. Soc. 63 (1967), 55–71. | MR | Zbl

[22] S. G. Vlăduts, Y. I. Manin, Linear codes and modular curves. Itogi Nauki i Tekhniki (1984), 209–257. | MR | Zbl

[23] F. Voloch, M. Zarzar, Algebraic geometric codes on surfaces. SMF Séminaires et congrès 21 (2009).

[24] M. Zarzar, Error-correcting codes on low rank surfaces. Finite Fields Appl. 13(4) (2007), 727–737. | MR | Zbl