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Since the proof in 1982, by Tsfasman Vlȃduţ 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 2 , 2 49, elements. The complexity of the algorithm, as measured in terms of multiplications and divisions over the finite field GF (2), is upper-bounded by [ log ()] 3 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-Varshamov (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

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Abstract. Extending results of Schindler [Sch] and Hamkins and Welch [HW03], we establish in the context of infinite time Turing machines that P is properly contained in NP ∩ co-NP. Furthermore, NP ∩ co-NP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P + = NP + ∩ co-NP + = NP ∩ co-NP, though P ++ is properly contained in NP ++ ∩ co-NP ++ . Within any contiguous block of infinite clockable ordinals, we show that P α = NP α ∩ co-NP α , but if β begins a gap in the clockable ordinals, then P β = NP β ∩ co-NP β . Finally, we establish that P f = NP f ∩ co-NP f for most functions f : R → ord, although we provide examples where P f = NP f ∩ co-NP f and P f = NP f .