Construction of algebraic error control codes (ECC) on the elliptic Riemann surface

Abstract

Summary form only given, as follows. The authors make use of the power series facility Scratchpad2, a symbolic computer language under development at the IBM Research Mathematics Department, to construct algebraic ECC on the elliptic Riemann surface of genus one. The construction method described uses products of elliptic theta series in one variable with broken characteristics, which define the projective coordinates of an elliptic curve. The curve equation is an identity satisfied by the power series. The points on the curve used to locate the symbols inside the block are division values of the elliptic periods. The linear transformation formulas of the elliptic theta series are used to compute the coordinates of these points. A Pade approximation algorithm produces rational approximants to quotients of these coordinates. The check matrix is obtained by evaluating an adjoint linear system of functions at these points. This computation is carried out with the Pade approximants. The nullspace of this matrix is the generator of the code. The authors described this construction by using a specific example that highlights the method.