Algebraic List Decoding of Elliptic Codes Through Module Basis Reduction

Elliptic codes is an important class of algebraic-geometric (AG) codes due to their least genus penalty. Their codeword length can exceed that of Reed-Solomon (RS) codes defined over the same finite field, resulting in a greater error-correction capability. This paper proposes the module basis reduction (BR) technique for solving the interpolation problem in algebraic list decoding (ALD) of one-point elliptic codes. A basis of the module that satisfies all interpolation constrains can be constructed by defining the explicit Lagrange interpolation function over the elliptic function field. They lead to the generators for the module basis. The basis can be further reduced to the desired Gröbner basis which contains the minimum interpolation polynomial Q(x, y, z). Compared with Koetter's interpolation, the BR interpolation technique significantly reduces the complexity in finding Q(x, y, z). Our analysis shows the BR interpolation complexity will reduce as the code rate increases.