8 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
Mathematical approach to channel codes with a diagonal matrix structure
Digital communications have now become a fundamental part of modern society. In communications,
channel coding is an effective way to reduce the information rate down to channel
capacity so that the information can be transmitted reliably through the channel. This thesis is
devoted to studying the mathematical theory and analysis of channel codes that possess a useful
diagonal structure in the parity-check and generator matrices. The first aspect of these codes
that is studied is the ability to describe the parity-check matrix of a code with sliding diagonal
structure using polynomials. Using this framework, an efficient new method is proposed to obtain
a generator matrix G from certain types of parity-check matrices with a so-called defective
cyclic block structure. By the nature of this method, G can also be completely described by a
polynomial, which leads to efficient encoder design using shift registers. In addition, there is no
need for the matrices to be in systematic form, thus avoiding the need for Gaussian elimination.
Following this work, we proceed to explore some of the properties of diagonally structured lowdensity
parity-check (LDPC) convolutional codes. LDPC convolutional codes have been shown
to be capable of achieving the same capacity-approaching performance as LDPC block codes
with iterative message-passing decoding. The first crucial property studied is the minimum
free distance of LDPC convolutional code ensembles, an important parameter contributing to
the error-correcting capability of the code. Here, asymptotic methods are used to form lower
bounds on the ratio of the free distance to constraint length for several ensembles of asymptotically
good, protograph-based LDPC convolutional codes. Further, it is shown that this ratio
of free distance to constraint length for such LDPC convolutional codes exceeds the ratio of
minimum distance to block length for corresponding LDPC block codes.
Another interesting property of these codes is the way in which the structure affects the performance
in the infamous error floor (which occurs at high signal to noise ratio) of the bit error
rate curve. It has been suggested that “near-codewords” may be a significant factor affecting
decoding failures of LDPC codes over an additive white Gaussian noise (AWGN) channel.
A near-codeword is a sequence that satisfies almost all of the check equations. These nearcodewords
can be associated with so-called ‘trapping sets’ that exist in the Tanner graph of a
code. In the final major contribution of the thesis, trapping sets of protograph-based LDPC convolutional
codes are analysed. Here, asymptotic methods are used to calculate a lower bound
for the trapping set growth rates for several ensembles of asymptotically good protograph-based
LDPC convolutional codes. This value can be used to predict where the error floor will occur
for these codes under iterative message-passing decoding
Split weight enumerators for the preparata codes with applications to designs
For quaternary Preparata and Kerdock codes of length N=2(m), m odd, we prove that the split complete weight enumerator for a coordinate partition into 3 and N-3 coordinates is independent of the chosen partition. The result implies that the words of a given complete weight in either a Preparata code or Kerdock code define a 3-design.X117sciescopu