On decoding up to error correcting capacity of linear error-correcting codes with Gröbner bases

The problem of decoding up to error correcting capacity of arbitrary linear codes with the use of Gröbner bases is addressed. A new method is proposed, which is based on reducing an initial decoding problem to solving some system of polynomial equations over a finite field. The peculiarity of this system is that, when we want to decode up to half the minimum distance, it has a unique solution even over the algebraic closure of the considered finite field, although field equations are not added. The equations in the system have degree at most 2. Some experimental results for the method are presented.