The thesis develops a new approach to the central themes of algebraic coding theory. The focus here is the newly introduced concept of a partial-inverse polynomial in a quotient ring F [x]/m(x). In particular, the decoding of Reed-Solomon codes can be attributed to the computation of a partial-inverse polynomial. The problem of practical computation of a partial-inverse polynomial is closely re...

Reed-Solomon (RS) codes are among the most ubiquitous codes due to their good parameters as well as efficient encoding and decoding procedures. However, RS codes suffer from having a fixed length. In many applications where the length is static, the appropriate length can be obtained by an RS code by shortening or puncturing. Generalized Reed-Solomon (GRS) codes are a generalization of RS codes...

We propose an improvement to Sudan's algorithm for decoding Reed-Solomon codes beyond half of their minimum distance, and its generalisation to algebraic-geometric codes. Both algorithms, in their original version, involve factorisation of polynomials. The main idea consists in replacing factorisation by an iterative root nding procedure of low complexity based on Newton approximation. In the c...

The astonishing performance of concatenated codes attracted many researchers and this has resulted in an explosive amount of literature since their introduction few years ago. As concatenation between codes did not actually arise from applying a pre-existing theory, most of their outstanding features remain to be explained. This paper gives some explanation of the concatenation between Reed Sol...

We present a generalisation of Twisted Reed– Solomon codes containing a new large class of MDS codes. We prove that the code class contains a large subfamily that is closed under duality. Furthermore, we study the Schur squares of the new codes and show that their dimension is often large. Using these structural properties, we single out a subfamily of the new codes which could be considered fo...

Long linear codes constructed from toric varieties over finite fields, their multiplicative structure and decoding. The main theme is the inherent multiplicative structure on toric codes. The multiplicative structure allows for decoding, resembling the decoding of Reed-Solomon codes and aligns with decoding by error correcting pairs. We have used the multiplicative structure on toric codes to c...

Error correcting codes are used for detection and correction of errors in digital communication system. Error correcting coding is based on appending of redundancy to the information message according to a prescribed algorithm. Reed Solomon codes are part of channel coding and withstand the effect of noise, interference and fading. Galois field arithmetic is used for encoding and decoding reed ...

It has been proved that the maximum likelihood decoding problem of Reed-Solomon codes is NP-hard. However, the length of the code in the proof is at most polylogarithmic in the size of the alphabet. For the complexity of maximum likelihood decoding of the primitive Reed-Solomon code, whose length is one less than the size of alphabet, the only known result states that it is at least as hard as ...

Reed-Solomon codes are a class of maximum distance separable error correcting codes with known fast error correction algorithms. They have been widely used to assure data integrity for stored data on compact discs, DVDs, and in RAID storage systems, for digital communications channels such as DSL internet connections, and for deep space communications on the Voyager mission. The recent explosio...

Symbolic q-ary multithreshold decoding (qMTD) of q-ary self-orthogonal codes is submitted. It’s shown the BER performance of qMTD is close to the results provided by optimum total search methods, which are not realizable for nonbinary codes in general. New decoders are compared with different decoders of Reed-Solomon codes. The performance provided by non-binary MTD in some cases is unattainabl...