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Differential approach for the study of duals of algebraic-geometric codes on surfaces. (English. French summary) Zbl 1278.14036

It is well known that the duals of algebraic geometry codes obtained from curves by evaluating functions from a Riemann–Roch space in rational points on the curve, can be described using differentials. However, dual codes obtained in a similar way from higher dimensional algebraic varieties, are not completely understood. In this article, the author manages to use differentials to describe these dual codes coming from surfaces. It is shown that in general the dual code can be obtained by taking the sum of several differential codes. Then the parameters of such codes are studied in case the surface has Picard number 1. Some of the codes obtained have parameters as good as the best known codes up to now.

MSC:

14G50 Applications to coding theory and cryptography of arithmetic geometry
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)

Software:

Code Tables; MinT
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References:

[1] Y. Aubry, Reed-Muller codes associated to projective algebraic varieties. Lecture Notes in Math. 1518 (1992), 4-17. · Zbl 0781.94004
[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. · Zbl 1116.14012
[3] A. Couvreur, Sums of residues on algebraic surfaces and application to coding theory. J. of Pure and Appl. Algebra 213 (2009), 2201-2223. · Zbl 1174.14023
[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. · Zbl 0267.94014
[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. · Zbl 1063.94107
[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. · Zbl 1155.94022
[8] F. A. B. Edoukou, Codes defined by forms of degree 2 on quadric surfaces. IEEE Trans. Inform. Theory 54(2) (2008), 860-864. · Zbl 1311.94117
[9] V. D. Goppa, Codes on algebraic curves. Dokl. Akad. Nauk SSSR. 259(6) (1981), 1289-1290. · Zbl 0489.94014
[10] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, (2007). Accessed on 2010-11-15, .
[11] R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics, 1977. · Zbl 0531.14001
[12] J. Kollàr, K. E. Smith, A. Corti, Rational and nearly rational varieties. Cambridge University Press, 2004. · Zbl 1060.14073
[13] F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. North-Holland Mathematical Library, 1977. · Zbl 0369.94008
[14] Martinínez-Moro, Edgar and Munuera, Carlos and Ruano, Diego, Advances in Algebraic Geometry Codes. World Scientific, 2008. · Zbl 1155.94006
[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. · Zbl 0358.14012
[16] V. S. Pless, W. C. Huffman, R. A. Brualdi, Handbook of coding theory. North-Holland Mathematical Library, 1998. · Zbl 0907.94001
[17] B. Poonen, Bertini theorems over finite fields. Ann. of Math. 160(3) (2004), 1099-1127. · Zbl 1084.14026
[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 . · Zbl 1130.65302
[19] J.-P. Serre, Lettre à M. Tsfasman. Astérisque (198-200) (1992), 351-353. · Zbl 0758.14008
[20] H. Stichtenoth, Algebraic function fields and codes. Universitext. Springer-Verlag, 1993. · Zbl 1155.14022
[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. · Zbl 0201.53702
[22] S. G. Vlăduts, Y. I. Manin, Linear codes and modular curves. Itogi Nauki i Tekhniki (1984), 209-257. · Zbl 0629.94013
[23] F. Voloch, M. Zarzar, Algebraic geometric codes on surfaces. SMF Séminaires et congrès 21 (2009). · Zbl 1216.94066
[24] M. Zarzar, Error-correcting codes on low rank surfaces. Finite Fields Appl.13(4) (2007), 727-737. · Zbl 1167.94008
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