On the key equation

We consider the set M={(a, b):a≡bh mod x^2t} 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