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Linkage and Codes on Complete Intersections

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract.

This note is meant to be an introduction to cohomological methods and their use in the theory of error-correcting codes. In particular we consider evaluation codes on a complete intersection. The dimension of the code is determined by the Koszul complex for X⊂ℙ2 and a lower bound for the minimal distance is obtained through linkage. By way of example our result fits the well-known formula for the minimal distance of the Generalized Reed-Muller code.

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References

  1. Auslander, M., Buchsbaum, D.A.: Codimension and multiplicity. Ann. of Math. 68(2), 625–657 (1958)

    MATH  Google Scholar 

  2. Auslander, M., Buchsbaum, D.A.: Corrections to Codimension and multiplicity. Ann. of Math. 70(2), 395–397 (1959)

    MATH  Google Scholar 

  3. Blake, I.F., Mullin, R.C.: The mathematical theory of coding. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, xi+356 1975

  4. Coppo, M.-A.: Une généralisation du théorème de Cayley-Bacharach. C. R. Acad. Sci. Paris Sér. I. Math. 314(8), 613–616 (1992)

    MATH  Google Scholar 

  5. Duursma I., Rentería, C., Tapia-Recillas, H., Reed-Muller codes on complete intersections. Applicable Algebra in Engineering, Communications and Computing. 11, 455–462 (2001)

    Google Scholar 

  6. Hansen, J.P.: Points in uniform position and maximum distance separable codes. Zero-dimensional schemes (Ravello, 1992), 205–211, Berlin: de Gruyter, 1994

    Google Scholar 

  7. Hartshorne, R.: Algebraic Geometry. New York: Springer 1977

  8. Iversen, B.: Cohomology of sheaves. Berlin: Springer xii+464, 1986

  9. Kasami, Tadao and Lin, Shu and Peterson, W. Wesley.: New generalizations of the Reed-Muller codes. I. Primitive codes. IEEE Trans. Information Theory. IT-14, 189–199 (1968)

    Google Scholar 

  10. Lachaud, G.: Number of points of plane sections and linear codes defined on algebraic varieties. Arithmetic, geometry and coding theory (Luminy, 1993), 77–104, Berlin: de Gruyter, 1996

    Google Scholar 

  11. Lax, R.F., Salazar, G.: An analoque of one-point codes for complete intersections varieties. preprint, 2000

  12. Lorenzini, A.: Betti numbers of points in projective space. Journal of Pure and Applied Algebra. 63(2), 181–193 (1990)

    Article  MATH  Google Scholar 

  13. Matsumura, H.: Commutative ring theory. Cambridge: Cambridge University Press, xiv+320 1989

  14. Matsumura, H.: Commutative algebra. Benjamin/Cummings Publishing Co., Inc., Reading, Mass. xv+313, 1980

  15. MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. II. North-Holland Mathematical Library, Vol. 16, Amsterdam: North-Holland Publishing Co. 1977

  16. Mercier D.-J., Rolland R.: Polynômes homogènes qui s’annulent sur l’espace projectif ℙm𝔽 q . Journal of Pure and Applied Algebra 124, 227–240 (1998)

    Article  Google Scholar 

  17. Peskine, C., Szpiro, L.: Liaison des variétés algébriques. I. Inv. Math. 26, 271–302 (1974)

    MATH  Google Scholar 

  18. Rentería, C., Tapia-Recillas, H.: Reed-Muller codes: an ideal theory approach. Comm. Algebra 25(2), 401–413 (1997)

    Google Scholar 

  19. Rentería, C., Tapia-Recillas, H.: Linear codes associated to the ideal of points in P d and its canonical module. Comm. Algebra 24(3), 1083–1090 (1996)

    Google Scholar 

  20. Sørensen, A.B.: Projective Reed-Muller codes. IEEE Trans. Information Theory 37(6), 1567–1576 (1991) 0018–9448

    Article  Google Scholar 

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Authors and Affiliations

  1. Matematisk Institut, Aarhus University, Ny Munkegade, 8000, Aarhus C, Denmark

    Johan P. Hansen

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  1. Johan P. Hansen
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Correspondence to Johan P. Hansen.

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Keywords: Liason, Linkage, Complete intersections, Error-correcting codes, Generalized Reed-Muller codes

Part of this work was done while visiting Institut de Mathématique de Luminy, 163 avenue de Luminy, Case 907, 13288 Marseille CEDEX 9, France

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Hansen, J. Linkage and Codes on Complete Intersections. AAECC 14, 175–185 (2003). https://doi.org/10.1007/s00200-003-0119-3

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  • DOI: https://doi.org/10.1007/s00200-003-0119-3

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