Journal ArticleDOI
Algebraic constructions of Shannon codes for regular channels
Philippe Delsarte,P. Piret +1 more
TLDR
A solution based on Justesen's idea of variable concatenated codes is given for the case of a symmetric memoryless channel with an input alphabet of prime power order, under the assumption that the information messages are equiprobable.Abstract:
The problem of the explicit construction of encoders achieving Shannon's capacity and admitting a simple decoding algorithm is considered. A solution based on Justesen's idea of variable concatenated codes is given for the case of a symmetric memoryless channel with an input alphabet of prime power order, under the assumption that the information messages are equiprobable. This construction remains good for a nonsymmetric channel provided the encoding rate is smaller than a well-defined "pseudocapacity." In case the channel is regular, it is shown that the error probability after decoding is an exponentially decreasing function of the block length for any encoding rate less than the channel capacity.read more
Citations
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Journal ArticleDOI
Good error-correcting codes based on very sparse matrices
TL;DR: It is proved that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit, and experimental results for binary-symmetric channels and Gaussian channels demonstrate that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved.
Dissertation
On the construction of some capacity-approaching coding schemes
Sae-Young Chung,G. David Forney +1 more
TL;DR: The reciprocal-channel approximation, based on dualizing LDPC codes, provides a very accurate model of density evolution for the AWGN channel, and another approximation method, Gaussian approximation, is developed, which enables us to visualize infinite-dimensional density evolution and optimization ofLDPC codes.
Journal ArticleDOI
Averaging bounds for lattices and linear codes
TL;DR: The relation between the combinatorial packing of solid bodies and the information-theoretic "soft packing" with arbitrarily small, but positive, overlap is illuminated and the "soft-packing" results are new.
Journal ArticleDOI
Exponential Decreasing Rate of Leaked Information in Universal Random Privacy Amplification
TL;DR: An exponential upper bound is derived for Eve's information in secret key generation from a common random number without communication based on the Rényi entropy of order 2 and is applied to secret key agreement by public discussion.
Journal ArticleDOI
Sphere-bound-achieving coset codes and multilevel coset codes
TL;DR: It is shown that the sphere bound can be approached by a large class ofcoset codes or multilevel coset codes with multistage decoding, including certain binary lattices, andExponential error bounds for coset code bounds are developed, generalizing Poltyrev's (1994) bounds for lattices.
References
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Journal ArticleDOI
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TL;DR: It is shown that i) erasures-and-errors decoding of Goppa codes can be done using O(n \log^{2} n) arithmetic operations, ii) long primitive binary Bose-Chaudhuri-Hocquenghem (BCH) codesCan be decoded using O-log n arithmetic Operations, and iii) Justesen's asymptotically good codes can been decoded use O( n^{2}) bit operations.
Journal ArticleDOI
Justesen's construction--The low-rate case (Corresp.)
TL;DR: By using the more general class of BCH codes, rather than RS codes, the results of Justesen [1] are improved for code rates below 0.07, and the value of d/n for the new codes approaches the Varsharmov-Gilbert hound.
Journal ArticleDOI
Certain generalizations of concatenated codes--Exponential error bounds and decoding complexity
TL;DR: New coding and decoding schemes based on concatenated codes are proposed, potentially superior to Forney's original concatenation scheme in the sense that for discrete memoryless channels the former has a smaller upper bound on the probability of decoding error for the same order of decoding complexity.