Bounded distance decoding of linear error-correcting codes with Gröbner bases

The problem of bounded distance decoding of arbitrary linear codes using Gröbner bases is addressed. A new method is proposed, which is based on reducing an initial decoding problem to solving a certain 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. As our experiments suggest, our method is much faster than the one of Fitzgerald–Lax. It is also shown via experiments that the proposed approach in some range of parameters is superior to the generic syndrome decoding.