Algebraic geometry codes from higher dimensional varieties

This paper is a general survey of literature on Goppa-type codes from higher dimensional algebraic varieties. The construction and several techniques for estimating the minimum distance are described first. Codes from various classes of varieties, including Hermitian hypersurfaces, Grassmannians, flag varieties, ruled surfaces over curves, and Deligne-Lusztig varieties are considered. Connections with the theories of toric codes and order domains are also briefly indicated.

[1]  Edoardo Ballico,et al.  The Horace Method for Error-Correcting Codes , 2006, Applicable Algebra in Engineering, Communication and Computing.

[2]  John Little,et al.  CAYLEY-BACHARACH AND EVALUATION CODES ON COMPLETE INTERSECTIONS , 2003, math/0311129.

[3]  Johan P. Hansen,et al.  Linkage and Codes on Complete Intersections , 2003, Applicable Algebra in Engineering, Communication and Computing.

[4]  J. H. Lint,et al.  Algebraic-geometric codes , 1992 .

[5]  F. Rodier.  Codes from Flag Varieties over a Finite Field , 2003 .

[6]  Johan P. Hansen,et al.  Toric Surfaces and Error-correcting Codes , 2000 .

[7]  Anders Bjært Sørensen.  Projective Reed-Muller codes , 1991, IEEE Trans. Inf. Theory.

[8]  Yves Aubry,et al.  Reed-Muller codes associated to projective algebraic varieties , 1992 .

[9]  Meinolf Geck,et al.  Finite groups of Lie type , 1985 .

[10]  Marcos Zarzar,et al.  Error-correcting codes on low rank surfaces , 2007, Finite Fields Their Appl..

[11]  Yves Aubry,et al.  Algebraic geometric codes on surfaces , 1992 .

[12]  Sudhir R. Ghorpade,et al.  Higher Weights of Grassmann Codes , 2000 .

[13]  Chris Lomont,et al.  Error Correcting Codes on Algebraic Surfaces , 2003, math/0309123.

[14]  Frédéric A. B. Edoukou.  Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture , 2007, Finite Fields Their Appl..

[15]  Gilles Lachaud,et al.  The parameters of projective Reed-Müller codes , 1990, Discret. Math..

[16]  P. Deligne.  La conjecture de Weil. I , 1974 .

[17]  S. Hansen,et al.  Canonical bundles of Deligne–Lusztig varieties , 1999 .

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  I︠u︡. I. Manin,et al.  Cubic forms; algebra, geometry, arithmetic , 1974 .

[20]  G. Lachaud.  Number of points of plane sections and linear codes defined on algebraic varieties , 1996 .

[21]  M. Boguslavsky,et al.  SECTIONS OF THE DEL PEZZO SURFACES AND GENERALIZED WEIGHTS , 2002 .

[22]  Rita Vincenti,et al.  On the Linear Codes Arising from Schubert Varieties , 2004, Des. Codes Cryptogr..

[23]  James W. P. Hirschfeld,et al.  The weight hierarchy of higher dimensional Hermitian codes , 1994, IEEE Trans. Inf. Theory.

[24]  Johan P. Hansen,et al.  Deligne-Lusztig varieties and group codes , 1992 .

[25]  Dmitry Yu. Nogin,et al.  Codes associated to Grassmannians , 1996 .

[26]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[27]  Andries E. Brouwer.  Linear spaces of quadrics and new good codes , 1998 .

[28]  S. Hansen,et al.  Picard Groups of Deligne-Lusztig Varieties - with a View toward Higher Codimensions , 2002 .

[29]  J. Brasselet.  Introduction to toric varieties , 2004 .

[30]  Alicia Urdapilleta,et al.  Generalized Hamming Weights for Linear Codes , 2001 .

[31]  Hans-Joachim Kroll,et al.  PD-sets for the codes related to some classical varieties , 2005, Discret. Math..

[32]  Ian F. Blake,et al.  Algebraic-Geometry Codes , 1998, IEEE Trans. Inf. Theory.

[33]  Zhe-xian Wan.  The weight hierarchies of the projective codes from nondegenerate quadrics , 1994, Des. Codes Cryptogr..

[34]  Ruud Pellikaan,et al.  Which linear codes are algebraic-geometric? , 1991, IEEE Trans. Inf. Theory.

[35]  Ruud Pellikaan,et al.  On the Structure of Order Domains , 2002 .

[36]  Ruud Pellikaan,et al.  Generalized Hamming Weights of q-ary Reed-Muller Codes , 1998, IEEE Trans. Inf. Theory.

[37]  S. Hansen,et al.  Error-Correcting Codes from Higher-Dimensional Varieties , 2001 .

[38]  A. Weil.  Numbers of solutions of equations in finite fields , 1949 .

[39]  Iwan M. Duursma,et al.  Reed-Muller Codes on Complete Intersections , 2001, Applicable Algebra in Engineering, Communication and Computing.

[40]  J. Thas,et al.  General Galois geometries , 1992 .

[41]  Jacques Wolfmann,et al.  Codes projectifs a deux ou trois poids associfs aux hyperquadriques d'une geometrie finie , 1975, Discret. Math..

[42]  Hans-Joachim Kroll,et al.  PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of PG(5, 2) , 2008, Discret. Math..

[43]  Michael A. Tsfasman,et al.  Schubert varieties, linear codes and enumerative combinatorics , 2005, Finite Fields Their Appl..

[44]  Frédéric A. B. Edoukou.  Codes Defined by Forms of Degree $2$ on Quadric Surfaces , 2008, IEEE Transactions on Information Theory.

[45]  Tohru Nakashima,et al.  Error-correcting codes on projective bundles , 2006, Finite Fields Their Appl..

[46]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[47]  John B. Little,et al.  The ubiquity of order domains for the construction of error control codes , 2003, Adv. Math. Commun..

[48]  F. Rodier,et al.  Nombre de points des surfaces de Deligne et Lusztig , 2000 .

[49]  I. Chakravarti.  Families of Codes with Few Distinct Weights from Singular and Non-Singular Hermitian Varieties and Quadrics in Projective Geometries and Hadamard Difference Sets and Designs Associated with Two-Weight Codes , 1990 .