A list decoding algorithm is presented for [ ] Reed-Solomon (RS) codes over GF ( ), which is capable of correcting more than ( ) 2 errors. Based on a previous work of Sudan, an extended key equation (EKE) is derived for RS codes, which reduces to the classical key equation when the number of errors is limited to ( ) 2 . Generalizing Massey's algorithm that finds the shortest recurrence that generates a given sequence, an algorithm is obtained for solving the EKE in time complexity ( ( ) ), where is a design parameter, typically a small constant, which is an upper bound on the size of the list of decoded codewords. (The case = 1 corresponds to classical decoding of up to ( ) 2 errors where the decoding ends with at most one codeword.) This improves on the time complexity ( ) needed for solving the equations of Sudan's algorithm by a naive Gaussian elimination. The polynomials found by solving the EKE are then used for reconstructing the codewords in time complexity (( log ) ( + log )) using root-finders of degree-univariate polynomials.

Techniques are presented for computing upper and lower bounds on the number of errors that can be corrected by list decoders for general block codes and, speci cally, for Reed-Solomon (RS) codes. The list decoder of Guruswami and Sudan implies such a lower bound (referred to here as the GS bound) for RS codes. It is shown that this lower bound, given by means of the code's length, the minimum Hamming distance, and the maximal allowed list size, applies in fact to all block codes. Ranges of code parameters are identi ed where the GS bound is tight for worst-case RS codes, in which case the list decoder of Guruswami and Sudan provably corrects the largest possible number of errors. On the other hand, ranges of parameters are provided for which the GS lower bound can be strictly improved. In some cases the improvement applies to all block codes with a given minimum Hamming distance, while in others it applies only to RS codes.

An input-constrained channel S is de ned as the set of words generated by a nite labeled directed graph. It is shown that every nite-state encoder with nite anticipation (i.e., with nite decoding delay) for S can beobtained through state-splitting rounds applied to some deterministic graph presentation of S , followed by a reduction of equivalent states. Furthermore, each splitting round can berestricted to follow a certain prescribed structure. This result, in turn, provides a necessary and su cient condition on the existence of nite-state encoders for S with a given rate p : q and a given anticipation a. A second condition is derived on the existence of such encoders this condition is only necessary, but it applies to every deterministic graph presentation of S. Based on these two conditions, lower bounds are derived on the anticipation of nite-state encoders. Those lower bounds improve on previously known bounds and, in particular, they are shown to be tight for the common rates used for the (1 7)-RLL and (2 7)-RLL constraints.

An input-constrained channel, or simply a constraint, is a set S of words that is generated by a nite labeled directed graph. An encoder for S maps in a lossless manner sequences of unconstrained input blocks into sequences of channel blocks, the latter sequences being words of S. In most applications, the encoders are nitestate machines and, thus, presented by state diagrams. In the special case where the state diagram of the encoder is (output) deterministic, only the current encoder state and the current channel block are needed for the decoding of the current input block. In this work, the problem of designing coding schemes that can serve two constraints simultaneously is considered. Speci cally, given two constraints S 1 and S 2 such that S 1 S 2 and two prescribed rates, conditions are provided for the existence of respective deterministic nite-state encoders E 1 and E 2 , at the given rates, such that (the state diagram of) E 1 is a subgraph of E 2. Such encoders are referred to as nested encoders. The provided conditions are also constructive in that they imply an algorithm for nding such encoders when they exist. The nesting structure allows to decode E 1 while using the decoder of E 2. Recent developments in optical recording suggest a potential application that can take a signi cant advantage of nested encoders.

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