239 research outputs found
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
Linear constructions for DNA codes
AbstractIn this paper we translate in terms of coding theory constraints that are used in designing DNA codes for use in DNA computing or as bar-codes in chemical libraries. We propose new constructions for DNA codes satisfying either a reverse-complement constraint, a GC-content constraint, or both, that are derived from additive and linear codes over four-letter alphabets. We focus in particular on codes over GF(4), and we construct new DNA codes that are in many cases better (sometimes far better) than previously known codes. We provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search
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-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups
Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology.
The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is q and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices BH(p, q)”, first discovered by A. T. Butson [6].
In this dissertation it is shown that an error correcting code whose codewords consist of real numbers in finite Galois field Gf( p) can be associated in a simple way with each Butson Hadamard matrix BH(p, q), where p \u3e 0 is a prime number. Distance properties of such codes are studied, as well as conditions for the existence of linear codes, for which standard decoding techniques are available.
In the search for cyclic linear generalized Hadamard codes, the concept of an M-invariant infinite sequence whose elements are integers in a finite field is introduced. Such sequences are periodic of least period, T, and have the interesting property, that arbitrary identical rearrangements of the elements in each period yields a periodic sequence with the same least period. A theorem characterizing such M-invariant sequences leads to discovery of a simple and efficient polynomial method for constructing generalized Hadamard matrices whose core is a linear cyclic matrix and whose row vectors constitute a linear cyclic error correcting code.
In addition, the problem is considered of determining parameter sequences {tn} for which the corresponding potential generalized Hadamard matrices BH(p, ptn) do not exist. By analyzing quadratic Diophantine equations, new methods for constructing such parameter sequences are obtained. These results show the rich number theoretic complexity of the existence question for generalized Hadamard matrices
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
Design of tch-type sequences for communications
This thesis deals with the design of a class of cyclic codes inspired by TCH codewords.
Since TCH codes are linked to finite fields the fundamental concepts and facts about abstract
algebra, namely group theory and number theory, constitute the first part of the thesis.
By exploring group geometric properties and identifying an equivalence between some operations
on codes and the symmetries of the dihedral group we were able to simplify the generation
of codewords thus saving on the necessary number of computations. Moreover, we
also presented an algebraic method to obtain binary generalized TCH codewords of length
N = 2k, k = 1,2, . . . , 16. By exploring Zech logarithm’s properties as well as a group theoretic
isomorphism we developed a method that is both faster and less complex than what was
proposed before. In addition, it is valid for all relevant cases relating the codeword length N
and not only those resulting from N = p
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