A low-complexity algorithm for the construction of algebraic-geometric codes better than the Gilbert-Varshamov bound

Abstract

Since the proof in 1982, by Tsfasman Vladut and Zink of the existence of algebraic-geometric (AG) codes with asymptotic performance exceeding the Gilbert-Varshamov (G-V) bound, one of the challenges in coding theory has been to provide explicit constructions for these codes. In a major step forward during 1995-1996, Garcia and Stichtenoth (GS) provided an explicit description of algebraic curves, such that AG codes constructed on them would have a performance better than the G-V bound. We present the first low-complexity algorithm for obtaining the generator matrix for AG codes on the curves of GS. The symbol alphabet of the AG code is the finite field of q²/sup, q² ≥ 49, elements. The complexity of the algorithm, as measured in terms of multiplications and divisions over the finite field GF(q²), is upper-bounded by [N log_q(N)]³ where N is the length of the code. An example of code construction using the above algorithm is presented. By concatenating the AG code with short binary block codes, it is possible to obtain binary codes with asymptotic performance close to the G-V bound. Some examples of such concatenation are included.