Enumeration of extended irreducible binary Goppa codes of degree 2m2^{m} and length 2n+12^{n}+1

Let $n$ be an odd prime and m>1 be a positive integer. We produce an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Some examples are given to illustrate our results

A STUDY OF LINEAR ERROR CORRECTING CODES

Since Shannon's ground-breaking work in 1948, there have been two main development streams of channel coding in approaching the limit of communication channels, namely classical coding theory which aims at designing codes with large minimum Hamming distance and probabilistic coding which places the emphasis on low complexity probabilistic decoding using long codes built from simple constituent codes. This work presents some further investigations in these two channel coding development streams. Low-density parity-check (LDPC) codes form a class of capacity-approaching codes with sparse parity-check matrix and low-complexity decoder Two novel methods of constructing algebraic binary LDPC codes are presented. These methods are based on the theory of cyclotomic cosets, idempotents and Mattson-Solomon polynomials, and are complementary to each other. The two methods generate in addition to some new cyclic iteratively decodable codes, the well-known Euclidean and projective geometry codes. Their extension to non binary fields is shown to be straightforward. These algebraic cyclic LDPC codes, for short block lengths, converge considerably well under iterative decoding. It is also shown that for some of these codes, maximum likelihood performance may be achieved by a modified belief propagation decoder which uses a different subset of 7^ codewords of the dual code for each iteration. Following a property of the revolving-door combination generator, multi-threaded minimum Hamming distance computation algorithms are developed. Using these algorithms, the previously unknown, minimum Hamming distance of the quadratic residue code for prime 199 has been evaluated. In addition, the highest minimum Hamming distance attainable by all binary cyclic codes of odd lengths from 129 to 189 has been determined, and as many as 901 new binary linear codes which have higher minimum Hamming distance than the previously considered best known linear code have been found. It is shown that by exploiting the structure of circulant matrices, the number of codewords required, to compute the minimum Hamming distance and the number of codewords of a given Hamming weight of binary double-circulant codes based on primes, may be reduced. A means of independently verifying the exhaustively computed number of codewords of a given Hamming weight of these double-circulant codes is developed and in coiyunction with this, it is proved that some published results are incorrect and the correct weight spectra are presented. Moreover, it is shown that it is possible to estimate the minimum Hamming distance of this family of prime-based double-circulant codes. It is shown that linear codes may be efficiently decoded using the incremental correlation Dorsch algorithm. By extending this algorithm, a list decoder is derived and a novel, CRC-less error detection mechanism that offers much better throughput and performance than the conventional ORG scheme is described. Using the same method it is shown that the performance of conventional CRC scheme may be considerably enhanced. Error detection is an integral part of an incremental redundancy communications system and it is shown that sequences of good error correction codes, suitable for use in incremental redundancy communications systems may be obtained using the Constructions X and XX. Examples are given and their performances presented in comparison to conventional CRC schemes

Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm

Asymmetric Encryption for Wiretap Channels

Since the definition of the wiretap channel by Wyner in 1975, there has been much research to investigate the communication security of this channel. This thesis presents some further investigations into the wiretap channel which improve the reliability of the communication security. The main results include the construction of best known equivocation codes which leads to an increase in the ambiguity of the wiretap channel by using different techniques based on syndrome coding. Best known codes (BKC) have been investigated, and two new design models which includes an inner code and outer code have been implemented. It is shown that best results are obtained when the outer code employs a syndrome coding scheme based on the (23; 12; 7) binary Golay code and the inner code employs the McEliece cryptosystem technique based on BKC0s. Three techniques of construction of best known equivocation codes (BEqC) for syndrome coding scheme are presented. Firstly, a code design technique to produce new (BEqC) codes which have better secrecy than the best error correcting codes is presented. Code examples (some 50 codes) are given for the case where the number of parity bits of the code is equal to 15. Secondly, a new code design technique is presented, which is based on the production of a new (BEqC) by adding two best columns to the parity check matrix(H) of a good (BEqC), [n; k] code. The highest minimum Hamming distance of a linear code is an important parameter which indicates the capability of detecting and correcting errors by the code. In general, (BEqC) have a respectable minimum Hamming distance, but are sometimes not as good as the best known codes with the same code parameters. This interesting point led to the production of a new code design technique which produces a (BEqC) code with the highest minimum Hamming distance for syndrome coding which has better secrecy than the corresponding (BKC). As many as 207 new best known equivocation codes which have the highest minimum distance have been found so far using this design technique.Ministry of Higher Education and Scientific Research, Kurdistan Regional Government, Erbil-Ira