On the key equation

We consider the set M = { ( a . 11): n bh mod.rL') of all solutions of the key equation for alternant codes, where h is the syndrome polynomial. In decoding these codes a particular solution (d. (T) E M is sought, subject to d and being relatively prime and satisfying certain degree conditions. We prove that these requirements specify (d. (T ) uniquely as the minimal element of M (analogous to the monic polynomial of minimal degree generating an ideal of F[.r]) with respect to a certain term order and that, as such, (d, (T ) 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-anderasures decoding) we derive a general algorithm for solving the congruence n = hg mod .r"' 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.