A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it retains the same advantages of fast speed, regular structure, and direct message recovery. Upon this generalization, we add a technique from the Guruswami-Sudan list decoding that boosts the decoding speed significantly. Compared with other known decoding algorithms for general AG codes, it has a similar decoding performance and allows streamlined practical implementation by its simple and regular structure.

We present a unique decoding algorithm of algebraic geometry (AG) codes on plane curves, Hermitian codes in particular, from an interpolation point of view. The algorithm successfully corrects errors of weight up to half of the order bound on the minimum distance of the AG code. It is the first decoding algorithm to combine some features of the interpolation-based list decoding with the performance of the syndrome decoding with the majority voting scheme. The regular structure of the algorithm allows a straightforward parallel implementation.

We study subfield-subcodes of Generalized Toric (GT) codes over Fps. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes. We identify polynomial generators for subfield-subcodes of GT codes which allows us to determine the dimensions and obtain bounds for the minimum distance. We give several examples of binary and ternary subfield-subcodes of GT codes that are the best known codes of a given dimension and length.

An algebraic soft-decision decoder for Hermitian codes is presented. We apply Koetter and Vardy's soft-decision decoding framework, now well established for Reed-Solmon codes, to Hermitian codes. First we provide an algebraic foundation for soft-decision decoding. Then we present an interpolation algorithm to find theQ-polynomial that plays a key role in the decoding. With some simulation results, we compare performances of the algebraic soft-decision decoders for Hermitian codes and Reed-Solmon codes, favorable to the former.

Hermitian codes offer desirable properties such as large code lengths, good error-correction at high code rates, etc. The main problem in making Hermitian codes practical is to find a way of performing the required computations in a fast and memory efficient way so as to satisfy resource and throughput constraints imposed by the systems. The paper presents some architecture for Hermitian decoders which enhance their applicability in communication systems. Formulae and architectures for gap detection and address generation unit for satisfying memory constraints have been presented, which amount to 50% savings in storage area and 10% savings in the number of clock cycles reported in literature. A semi-parallel architecture is proposed as a solution to the latency and resource requirements tradeoff, which improves the throughput about q times compared to the word-serial architecture at an expense of some q times more adders, multipliers and simple multiplexers, where the code is defined over GF(q2). For a t error correcting code, the resource load of the parallel architectures is about gamma(t/q + (q-3)/4)( t/q + (q-3)/4 + 1) times this architecture, where gamma is the resource requirement ratio of a multiplier and an inverter

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers.

Garcia and Stichtenoth discovered a tower of function fields that meets the Drinfeld-Vladut bound on the ratio of the number of points to the genus. For this tower, Pellikaan, Stichtenoth, and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this correspondence, a nonrecursive description of the semigroups is given and from this the enumeration of each of the semigroups is derived as well as its inverse. This enables us to find an explicit formula for the order (Feng-Rao) bound on the minimum distance of the associated one-point codes.

A central problem of algebraic soft-decision decoding of Reed-Solomon codes is to find the minimal polynomial of the ideal of interpolating polynomials with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Gröbner bases of modules.

A new method is given to construct low-density parity check codes the graphs of which are of designed girth. We give examples to illustrate the new method, and also present performance diagrams that suggest that these codes are as good as random codes in low SNR, and preferable to random codes at higher SNR.

This paper presents a generalization of the key equation to Hermitian codes. The syndrome is interpreted as a power series and the product of this power series with a locator polynomial gives the error evaluator polynomial. The computation of the evaluator polynomial may be done iteratively using a modified version of a previously published computationally efficient algorithm for computing locator polynomials.