227 research outputs found
Simultaneous Code/Error-Trellis Reduction for Convolutional Codes Using Shifted Code/Error-Subsequences
In this paper, we show that the code-trellis and the error-trellis for a
convolutional code can be reduced simultaneously, if reduction is possible.
Assume that the error-trellis can be reduced using shifted error-subsequences.
In this case, if the identical shifts occur in the subsequences of each code
path, then the code-trellis can also be reduced. First, we obtain pairs of
transformations which generate the identical shifts both in the subsequences of
the code-path and in those of the error-path. Next, by applying these
transformations to the generator matrix and the parity-check matrix, we show
that reduction of these matrices is accomplished simultaneously, if it is
possible. Moreover, it is shown that the two associated trellises are also
reduced simultaneously.Comment: 5 pages, submitted to the 2011 IEEE International Symposium on
Information Theor
Trellises for stabilizer codes: definition and uses
Trellises play an important theoretical and practical role for classical
codes. Their main utility is to devise complexity-efficient error estimation
algorithms. Here, we describe trellis representations for quantum stabilizer
codes. We show that they share the same properties as their classical analogs.
In particular, for any stabilizer code it is possible to find a minimal trellis
representation. Our construction is illustrated by two fundamental error
estimation algorithms.Comment: 5 pages, 2 figure
The trellis complexity of convolutional codes
Convolutional codes have a natural, regular, trellis structure that facilitates the implementation of Viterbi's algorithm. Linear block codes also have a natural, though not in general a regular, “minimal” trellis structure, which allows them to be decoded with a Viterbi-like algorithm. In both cases, the complexity of an unenhanced Viterbi decoding algorithm can be accurately estimated by the number of trellis edge symbols per encoded bit. It would therefore appear that we are in a good position to make a fair comparison of the Viterbi decoding complexity of block and convolutional codes. Unfortunately, however, this comparison is somewhat muddled by the fact that some convolutional codes, the punctured convolutional codes, are known to have trellis representations which are significantly less complex than the conventional trellis. In other words, the conventional trellis representation for a convolutional code may not be the “minimal” trellis representation. Thus ironically, we seem to know more about the minimal trellis representation for block than for convolutional codes. We provide a remedy, by developing a theory of minimal trellises for convolutional codes. This allows us to make a direct performance-complexity comparison for block and convolutional codes. A by-product of our work is an algorithm for choosing, from among all generator matrices for a given convolutional code, what we call a trellis-canonical generator matrix, from which the minimal trellis for the code can be directly constructed. Another by-product is that in the new theory, punctured convolutional codes no longer appear as a special class, but simply as high-rate convolutional codes whose trellis complexity is unexpectedly small
Approximate MAP Decoding on Tail-Biting Trellises
We propose two approximate algorithms for MAP decoding on tail-biting
trellises. The algorithms work on a subset of nodes of the tail-biting trellis,
judiciously selected. We report the results of simulations on an AWGN channel
using the approximate algorithms on tail-biting trellises for the
Extended Golay Code and a rate 1/2 convolutional code with memory 6.Comment: 5 pages, 2 figures, ISIT 200
Approximate Linear Time ML Decoding on Tail-Biting Trellises in Two Rounds
A linear time approximate maximum likelihood decoding algorithm on
tail-biting trellises is prsented, that requires exactly two rounds on the
trellis. This is an adaptation of an algorithm proposed earlier with the
advantage that it reduces the time complexity from O(mlogm) to O(m) where m is
the number of nodes in the tail-biting trellis. A necessary condition for the
output of the algorithm to differ from the output of the ideal ML decoder is
reduced and simulation results on an AWGN channel using tail-biting rrellises
for two rate 1/2 convoluational codes with memory 4 and 6 respectively are
reporte
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