Decoding Affine Variety Codes Using Gröbner Bases
Designs, Codes and Cryptography
A Systolic Array Implementation of the Feng-Rao Algorithm
IEEE Transactions on Computers
Finding recursions for multidimensional arrays
Information and Computation
Computing a Basis of L(D) on an Affine Algebraic Curve with One Rational Place at Infinity
AAECC-13 Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Computer algebra handbook
Good expander graphs and expander codes: parameters and decoding
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Multidimensional Systems and Signal Processing
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Multidimensional cyclic codes and Artin--Schreier type hypersurfaces over finite fields
Finite Fields and Their Applications
On the Structure of Order Domains
Finite Fields and Their Applications
Integral closures and weight functions over finite fields
Finite Fields and Their Applications
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It is proved that any algebraic-geometric (AG) code can be expressed as a cross section of an extended multidimensional cyclic code. Both AG codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets: AG codes are defined over the points of an algebraic curve, and an m-dimensional cyclic code is defined over the points in m-dimensional space. The power of the unified theory is in its description of decoding techniques using Grobner bases. In order to fit an AG code into this theory, a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position. For curves in special position, all computations can be performed with polynomials and this also makes it possible to use the theory of Grobner bases. Next, a transform is defined for AG codes which generalizes the discrete Fourier transform. The transform is also related to a Grobner basis, and is useful in setting up the decoding problem. In the decoding problem, a key step is finding a Grobner basis for an error locator ideal. For AG codes, multidimensional cyclic codes, and indeed, any cross section of an extended multidimensional cyclic code, Sakata's algorithm can be used to find linear recursion relations which hold on the syndrome array. In this general context, the authors give a self-contained and simplified presentation of Sakata's algorithm, and present a general framework for decoding algorithms for this family of codes, in which the use of Sakata's algorithm is supplemented by a procedure for extending the syndrome array