230 research outputs found
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
On the Shape of the General Error Locator Polynomial for Cyclic Codes
General error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes. We now present a conjecture on their sparsity, which would imply polynomial-time decoding for all cyclic codes. A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for specific code families, providing evidence to our conjecture. From these, a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with t <= 2 and n < 105, as well as for t = 3 and n < 63, except for some cases where the conjectured sparsity is proved by a computer check. Moreover, we summarize all related results, previously published, and we show how they provide further evidence to our conjecture. Finally, we discuss the link between our conjecture and the complexity of bounded-distance decoding of the cyclic codes
The Weight Distributions of a Class of Cyclic Codes with Three Nonzeros over F3
Cyclic codes have efficient encoding and decoding algorithms. The decoding
error probability and the undetected error probability are usually bounded by
or given from the weight distributions of the codes. Most researches are about
the determination of the weight distributions of cyclic codes with few
nonzeros, by using quadratic form and exponential sum but limited to low
moments. In this paper, we focus on the application of higher moments of the
exponential sum to determine the weight distributions of a class of ternary
cyclic codes with three nonzeros, combining with not only quadratic form but
also MacWilliams' identities. Another application of this paper is to emphasize
the computer algebra system Magma for the investigation of the higher moments.
In the end, the result is verified by one example using Matlab.Comment: 10 pages, 3 table
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
On Decoding of Quadratic Residue Codes
A binary Quadratic Residue(QR) code of length n is an (n, (n+1)/2) cyclic code over GF(2m) with generator polynomial g(x) where m is some integer. The length of this code is a prime number of the form n = 8l + 1 where l is some integer. The generator polynomial g(x) is defined by g(x)=∏_(i∈Q_n) (x-βi ) where β is a primitive nth root of unity in the finite field GF(2m) with m being the smallest positive integer such that n|2m-1 and Qn is the collection of all nonzero quadratic residues modulo n given by Qn={i│i≡j2 mod n for 1≤j≤n-1}. Algebraic approaches to the decoding of the quadratic residue (QR) codes were studied in [2], [3], [4], [5], [6] and [13]. Here, in this thesis, some new more general properties are found for the syndromes of the subclass of binary QR codes of length n = 8m + 1 or n = 8m - 1. A new algebraic decoding algorithm for the (41, 21, 9) binary QR code is presented by having the unknown syndrome S3 which is a necessary condition for decoding the (41, 21, 9) QR code
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