Algebraic Chase Decoding of Elliptic Codes Through Computing the Gröbner Basis

This paper proposes two interpolation-based algebraic Chase decoding for elliptic codes. It is introduced from the perspective of computing the Gröbner basis of the interpolation module, for which two Chase interpolation approaches are utilized. They are Kötter’s interpolation and the basis reduction (BR) interpolation. By identifying η unreliable symbols, 2η decoding test-vectors are formulated, and the corresponding interpolation modules can be defined. The re-encoding further helps transform the test-vectors, facilitating the two interpolation techniques. In particular, Kötter’s interpolation is performed for the common elements of the test-vectors, producing an intermediate outcome that is shared by the decoding of all test-vectors. The desired Gröbner bases w.r.t. all test-vectors can be obtained in a binary tree growing fashion, leading to a low complexity but its decoding latency cannot be contained. In contrast, the BR interpolation first performs the common computation in basis construction which is shared by all interpolation modules, and then conducts the module basis construction and reduction for all test-vectors in parallel. It results in a significantly lower decoding latency. Finally, simulation results are also presented to demonstrate the effectiveness of the proposed Chase decoding.