2,892 research outputs found
Numerical and analytical bounds on threshold error rates for hypergraph-product codes
We study analytically and numerically decoding properties of finite rate
hypergraph-product quantum LDPC codes obtained from random (3,4)-regular
Gallager codes, with a simple model of independent X and Z errors. Several
non-trival lower and upper bounds for the decodable region are constructed
analytically by analyzing the properties of the homological difference, equal
minus the logarithm of the maximum-likelihood decoding probability for a given
syndrome. Numerical results include an upper bound for the decodable region
from specific heat calculations in associated Ising models, and a minimum
weight decoding threshold of approximately 7%.Comment: 14 pages, 5 figure
Coding for Parallel Channels: Gallager Bounds for Binary Linear Codes with Applications to Repeat-Accumulate Codes and Variations
This paper is focused on the performance analysis of binary linear block
codes (or ensembles) whose transmission takes place over independent and
memoryless parallel channels. New upper bounds on the maximum-likelihood (ML)
decoding error probability are derived. These bounds are applied to various
ensembles of turbo-like codes, focusing especially on repeat-accumulate codes
and their recent variations which possess low encoding and decoding complexity
and exhibit remarkable performance under iterative decoding. The framework of
the second version of the Duman and Salehi (DS2) bounds is generalized to the
case of parallel channels, along with the derivation of their optimized tilting
measures. The connection between the generalized DS2 and the 1961 Gallager
bounds, addressed by Divsalar and by Sason and Shamai for a single channel, is
explored in the case of an arbitrary number of independent parallel channels.
The generalization of the DS2 bound for parallel channels enables to re-derive
specific bounds which were originally derived by Liu et al. as special cases of
the Gallager bound. In the asymptotic case where we let the block length tend
to infinity, the new bounds are used to obtain improved inner bounds on the
attainable channel regions under ML decoding. The tightness of the new bounds
for independent parallel channels is exemplified for structured ensembles of
turbo-like codes. The improved bounds with their optimized tilting measures
show, irrespectively of the block length of the codes, an improvement over the
union bound and other previously reported bounds for independent parallel
channels; this improvement is especially pronounced for moderate to large block
lengths.Comment: Submitted to IEEE Trans. on Information Theory, June 2006 (57 pages,
9 figures
Application of Computer Algebra in List Decoding
The amount of data that we use in everyday life (social media, stock analysis, satellite communication etc.) are increasing day by day. As a result, the amount of data needs to be traverse through electronic media as well as to store are rapidly growing and there exist several environmental effects that can damage these important data during travelling or while in storage devices. To recover correct information from noisy data, we do use error correcting codes. The most challenging work in this area is to have a decoding algorithm that can decode the code quite fast, in addition with the existence of the code that can tolerate highest amount of noise, so that we can have it in practice.
List decoding is an active research area for last two decades. This research popularise in coding theory after the breakthrough work by Madhu Sudan where he used list decoding technique to correct errors that exceeds half the minimum distance of Reed Solomon codes. Towards the direction of code development that can reach theoretical limit of error correction, Guruswami-Rudra introduced folded Reed Solomon codes that reached at To decode this codes, one has to first interpolate a multivariate polynomial first and then have to factor out all possible roots. The difficulties that lies here are efficient interpolation, dealing with multiplicities smartly and efficient factoring. This thesis deals with all these cases in order to have folded Reed Solomon codes in practice
Asymptotic Analysis of SU-MIMO Channels With Transmitter Noise and Mismatched Joint Decoding
Hardware impairments in radio-frequency components of a wireless system cause
unavoidable distortions to transmission that are not captured by the
conventional linear channel model. In this paper, a 'binoisy' single-user
multiple-input multiple-output (SU-MIMO) relation is considered where the
additional distortions are modeled via an additive noise term at the transmit
side. Through this extended SU-MIMO channel model, the effects of transceiver
hardware impairments on the achievable rate of multi-antenna point-to-point
systems are studied. Channel input distributions encompassing practical
discrete modulation schemes, such as, QAM and PSK, as well as Gaussian
signaling are covered. In addition, the impact of mismatched detection and
decoding when the receiver has insufficient information about the
non-idealities is investigated. The numerical results show that for realistic
system parameters, the effects of transmit-side noise and mismatched decoding
become significant only at high modulation orders.Comment: 16 pages, 7 figure
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