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

Code construction on modular curves

Cataloged from PDF version of article.In this thesis, we have introduced two approaches on code construction on modular curves and stated the problems step by step. Moreover, we have given solutions of some problems in road map of code construction. One of the approaches uses mostly geometric and algebraic tools. This approach studies local invariants of the plane model Z0(`) of the modular curve Y0(`) given by the modular equation Φ` in affine coordinates. The approach is based on describing the hyperplane of regular differentials of Z0(`) vanishing at a given Fp 2 rational point. As constructing a basis for the regular differentials of Z0(`), we need to investigate its singularities. We have described the singularities of Z0(`) for prime ` in both characteristic 0 and positive characteristic. We have shown that all singularities of of the affine part, Z0(`), are self intersections. These self intersections are all simple nodes in characteristic 0 whereas the order of contact of any two smooth branches passing though a singular point may be arbitrarily large in characteristic p > 3 where p 6= `. Moreover the self intersections in characteristic zero are double. Indeed, structure of singularities of the affine curve Z0(`) essentially depends on two types of elliptic curves: The singularities corresponding to ordinary elliptic curves and the singularities corresponding to supersingular elliptic curves. The singularities corresponding to ordinary elliptic curves are all double points even though they are not necessarily simple nodes as in the case of characteristic 0. The singularities corresponding to supersingular elliptic curves are the most complicated ones and it may happen that there are more then two smooth branches passing though such kind of a singular point. We have computed the order of contact of any two smooth branches passing though a singular point both for ordinary case and for supersingular case.We have also proved that two points of Z0(`) at ∞ are cusps for odd prime ` which are analytically equivalent to the cusp of 0, given by the equation x ` = y `−1 . These two cusps are permuted by Atkin-Lehner involution. The multiplicity of singularity of each cusp is (`−1)(`−2) 2 . This result is valid in any characteristic p 6= 2, 3. The second approach is based on describing the Goppa codes on modular curve Y (`) as P SL2(F`) module. The main problem in this approach is investigating the structure of a group code as P SL2(F`) module. We propose a way of computing the characters of representations of a group code by using the localization formula. Moreover, we give an example of computing the characters of the code which associated to a canonical divisor on Y (`).Kara, OrhunPh.D

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