IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 6, SEPTEMBER 2001 2225 A Low-Complexity Algorithm for the Construction of Algebraic-Geometric Codes Better Than the Gilbert–Varshamov Bound Kenneth W. Shum, Student Member, IEEE, Ilia Aleshnikov, P. Vijay Kumar, Senior Member, IEEE, Henning Stichtenoth, and Vinay Deolalikar Abstract—Since the proof in 1982, by Tsfasman Vl˘ adut¸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 (G–S) provided an explicit description of algebraic curves, such that AG codes constructed on them would have performance better than the G–V bound. We present here the first low-complexity algorithm for obtaining the generator matrix for AG codes on the curves of G–S. The symbol alphabet of the AG code is the finite field of , , elements. The complexity of the algorithm, as measured in terms of multiplications and divisions over the finite field GF , is upper-bounded by where 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. Index Terms—Algebraic-geometric (AG) codes, concatenated codes, function field tower, Gilbert–Varshamov (G–V) bound. I. INTRODUCTION L ONG codes are judged on the basis of their parameters where is the relative minimum distance and is the code rate, i.e., if are the length, dimension and minimum distance of the code, respectively, then and The best long codes lie in the region defined by the Gilbert–Var- shamov (G–V) lower bound and the McEliece, Rodemich, Rumsey, and Welch [1] upper bound (see Fig. 1). One of the challenges in coding theory has been the construction of codes with symbol alphabet size fixed at and growing length whose Manuscript received November 6, 2000; revised March 2, 2001. This work was supported by the National Science Foundation under Grant CCR-0073555. K. W. Shum and P. V. Kumar are with the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089-2565 USA (e-mail: keshum@milly.usc.edu; vijayk@ceng.usc.edu). I. Aleshnikov and H. Stichtenoth are with Mathematik und Informatik, Universität GH Essen, Fachbereich 6, D-45117 Essen, Germany (e-mail: mat314@uni-essen.de; stichtenoth@uni-essen.de). V. Deolalikar is with the Information Theory Research Group, Hewlett- Packard Research Laboratories, Palo Alto, CA 94306 USA (e-mail: vinayd@ exch.hpl.hp.com). Communicated by J. Justesen, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(01)05463-3. Fig. 1. Upper and lower bound for asymptotic code parameters over GF . performance exceeds that of the G–V bound. It is desirable to keep small as this allows for simpler encoding and decoding. While it is known that there exist long binary alternant and concatenated codes that meet the G–V bound, no explicit description of these codes exists. It is an open question as to whether there exist long binary codes with performance im- proving upon the G–V bound. A similar statement was true in the nonbinary case until the early 1980s. Around 1980, V. D. Goppa [2] used the theory of algebraic curves to construct a new family of codes, now referred to as algebraic-geometric (AG) codes. The length of an AG code defined over a curve of genus , is roughly equal to the number of rational points on the curve, i.e., equal to the number of points having coordinates in the finite field of elements over which the curve is defined. The performance of an AG code of length is governed by the equation and thus depends upon the ratio . Good codes result when the ratio is small. However, the Drinfeld–Vl˘ adut ¸ (D–V) bound states that this ratio cannot be too small 0018–9448/01$10.00 © 2001 IEEE