A Generalized Rational Interpolation Problem and the Solution of theWelch–Berlekamp Key Equation
Designs, Codes and Cryptography
Lifting Decoding Schemes over a Galois Ring
AAECC-14 Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
A Unifying System-Theoretic Framework for Errors-and-Erasures Reed-Solomon Decoding
AAECC-14 Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
On the Gröbner bases of some symmetric systems and their application to coding theory
Journal of Symbolic Computation
List decoding of Reed-Solomon codes from a Gröbner basis perspective
Journal of Symbolic Computation
Key equations for list decoding of Reed-Solomon codes and how to solve them
Journal of Symbolic Computation
Algebraic decoding of negacyclic codes over $${\mathbb Z_4}$$
Designs, Codes and Cryptography
Hi-index | 754.84 |
We consider the set M={(a, b):a≡bh mod x2t} of all solutions of the key equation for alternant codes, where h is the syndrome polynomial. In decoding these codes a particular solution (ω, σ)∈M is sought, subject to ω and σ being relatively prime and satisfying certain degree conditions. We prove that these requirements specify (ω, σ) uniquely as the minimal element of M (analogous to the monic polynomial of minimal degree generating an ideal of F[x]) with respect to a certain term order and that, as such, (ω, σ) may be determined from an appropriate Grobner basis of M. Motivated by this and other variations of the key equation (such as that appropriate to errors-and-erasures decoding) we derive a general algorithm for solving the congruence a≡bg mod xn for a range of term orders defined by the conditions on the particular solution required. Our techniques provide a unified approach to the solution of these key equations