Generalized Inverse Based Decoding

The concept of Generalized Inverse Decoding (GID) is introduced, as an algebraic framework for the syndrome decoding (SD) and low-weight codeword (LWC) problems. The framework has ground on two characterizations by generalized inverses, one for the null space of a matrix and the other for the solution space of a system of linear equations over a finite field. Generic GID solvers are proposed for the SD and LWC problems. It is shown that information set decoding (ISD) algorithms, such as Prange, Lee-Brickell, Leon, and Stern’s algorithms, are particular cases of GID solvers. All of them search generalized inverses or elements of the null space under various specific strategies. However, as the paper shows in the case of Prange’s algorithm, they do not search through the entire space, while our solvers do even when they use just one Gaussian elimination. Apart from these, our GID framework clearly shows how each ISD algorithm except for Prange’s can be used as an SD or LWC solver. Experimental results show a very good behavior of the GID solvers. The domain of easy weights can be reached by a very few iterations and even enlarged.