This paper proposes the algebraic soft decoding (ASD) for one-point elliptic codes, where the interpolation problem is solved from the perspective of module basis reduction. In ASD, the interpolation polynomial… Click to show full abstract
This paper proposes the algebraic soft decoding (ASD) for one-point elliptic codes, where the interpolation problem is solved from the perspective of module basis reduction. In ASD, the interpolation polynomial $\mathcal {Q}(x, y, z)$ is the minimum candidate of a Gröbner basis. Based on a multiplicity matrix, an interpolation ideal can be defined. With the decoding output list size, an equivalent interpolation module can be led to. By further defining the set of interpolation points, a sequence of modules from the elliptic curve coordinate ring can be obtained. Based on the Lagrange interpolation functions over elliptic function field, a basis of the interpolation module can be constructed. The desired Gröbner basis that contains $\mathcal {Q}$ can be determined by reducing the module basis. Re-encoding transform (ReT) is further introduced to reduce the basis reduction complexity. It is also shown that the interpolation can be facilitated by assessing the degree of the Lagrange interpolation polynomials. The decoding complexity is analyzed, which is verified by numerical results. That shows the advantage of this interpolation technique over the conventional Kötter’s interpolation. The ASD performance of elliptic codes is also presented.
