19 research outputs found
Good Codes From Generalised Algebraic Geometry Codes
Algebraic geometry codes or Goppa codes are defined with places of degree
one. In constructing generalised algebraic geometry codes places of higher
degree are used. In this paper we present 41 new codes over GF(16) which
improve on the best known codes of the same length and rate. The construction
method uses places of small degree with a technique originally published over
10 years ago for the construction of generalised algebraic geometry codes.Comment: 3 pages, to be presented at the IEEE Symposium on Information Theory
(ISIT 2010) in Austin, Texas, June 201
Generating generalized necklaces and new quasi-cyclic codes
In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent polynomials from which defining polynomials for constructing quasi-cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13)
New Linear Codes over , , and
Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets , , and with better parameters. A total of 38 new linear codes over the three alphabets are presented
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
Z_q(Z_q+uZ_q)-Linear Skew Constacyclic Codes
In this paper, we study skew constacyclic codes over the ring where , for a prime and We give the definition of these codes as subsets of the ring . Some structural properties of the skew polynomial ring are discussed, where is an automorphism of We describe the generator polynomials of skew constacyclic codes over also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over we obtained some new linear codes over . Finally, we have generalized these codes to double skew constacyclic codes over