New lower bounds on the covering radius for different subclasses of irreducible Goppa codes are obtained.

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It is shown that for some location sets and some integral multiple power Galois fields that some new general subclasses of Goppa codes may be defined which have improved lower bounds to code dimension and minimum distance compared with ordinary Goppa codes. Some previously published results are shown to be particular cases of these general subclasses of codes. A new subclass of reversible Goppa codes is also presented. Examples of code construction for these subclasses, demonstrating the improved code parameters, are presented for both non binary and binary codes.

We present a generalized construction of extended length Goppa codes. Using this construction, we obtain 71 new codes in finite field Fqfor q = 4, 7, 8, 9 with better minimum distance than the previously known codes with the same length and dimension.

Two generalizations of the Hartmann-Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved. Our technique of embedding the code into a cyclic product code can be applied to other bounds, too and therefore generalizes them.

Goppa codes have some of the largest minimum distances possible for linear codes. The authors use some binary Goppa codes in which four new binary codes are obtained with parameters better than any codes known to date. The authors also present the necessary conditions for a code obtained by shortening a longer code which will have a greater minimum distance than the original code.

The subclass of cyclic Goppa codes is proposed. It is proved that this subclass contains all cyclic codes of length n|qm±1 with generator polynomial g(x), g(αi)=g(α-j)=0, i=0,1,..., s1, j=1,..., s2, αn=1, α ∈ GF(q2m).

Known properties of cyclic codes are used to give a unified description of many classical decoding algorithms for Reed-Solomon codes up to half the minimum distance. This description allows also simplified proofs for these decoders. Further, a novel decoding algorithm is derived using these properties directly and variants of a new error/erasure decoding algorithm are given. For decoding beyond half the minimum distance, a basis of all solutions for decoding is derived. This basis allows to use side information in order to decode beyond half the minimum distance. Other methods where this basis can be used are power decoding, also known as virtual syndrome extension, where additional equations are generated by taking powers of the received symbols, and interleaved Reed-Solomon codes. The extended Euclidean algorithm, which calculates the greatest common divisor, plays an essential role in many presented methods.

We present an efficient construction of extended length Goppa codes. Using this construction, we obtain 78 new nonbinary codes with better minimum distance than the previously known codes with the same length and dimension. The construction is based on the observation that certain Goppa codes can be seen as BCH codes.

A new approach to bound the minimum distance of q-ary cyclic codes is presented. The connection to the BCH and the Hartmann-Tzeng bound is formulated and it is shown that for several cases an improvement is achieved. We associate a second cyclic code to the original one and bound its minimum distance in terms of parameters of the associated code.

A new lower bound on the minimum distance ofq-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem bound and, for some codes, upon the Hartmann-Tzeng bound. Several Boston bounds are special cases of our bound. For some classes of codes, the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.

A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum distance is proven and several classes of binary cyclic codes are identified. For some classes of codes, this bound is better than the known bounds (e.g. BCH or Hartmann-Tzeng bound). Furthermore, a quadratic-time decoding algorithm up to this new bound is developed.

A novel method for joint decoding of interleaved Reed-Solomon (RS) codes is presented which is based on an extension of the Euclidean algorithm. This method differs from the known multi-sequence shift register algorithms which is shown by an example.

It is shown that subclasses of separable binary Goppa codes,Gamma(L,G) - codes, withL={alpha isin GF(22l):G(alpha) ne 0 } and special Goppa polynomialsG(x) can be presented as a chain of embedded and equivalent codes. The true minimal distance has been obtained for all codes of the chain.

A subclass of binary Goppa codes specified by a separable polynomial G(x)=x/sup t/+A and a subset L of elements of GF(2/sup m/) (no element of L may be a root of G(x) and t|(2/sup m/-1), A is a tth power in {GF(2/sup m/)/{0}}, is studied. For such codes it is shown that their minimal distance is equal to the design distance d=2t+1.>