29 research outputs found
Compact QC-LDPC Block and SC-LDPC Convolutional Codes for Low-Latency Communications
Low decoding latency and complexity are two important requirements of channel
codes used in many applications, like machine-to-machine communications. In
this paper, we show how these requirements can be fulfilled by using some
special quasi-cyclic low-density parity-check block codes and spatially coupled
low-density parity-check convolutional codes that we denote as compact. They
are defined by parity-check matrices designed according to a recent approach
based on sequentially multiplied columns. This method allows obtaining codes
with girth up to 12. Many numerical examples of practical codes are provided.Comment: 5 pages, 1 figure, presented at IEEE PIMRC 201
Hierarchical and High-Girth QC LDPC Codes
We present a general approach to designing capacity-approaching high-girth
low-density parity-check (LDPC) codes that are friendly to hardware
implementation. Our methodology starts by defining a new class of
"hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of
quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC
codes are composed of circulant sub-matrices, those of HQC LDPC codes are
composed of a hierarchy of circulant sub-matrices that are in turn constructed
from circulant sub-matrices, and so on, through some number of levels. We show
how to map any class of codes defined using a protograph into a family of HQC
LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the
degrees of freedom within the family of codes to yield a high-girth HQC LDPC
code. Finally, we discuss how certain characteristics of a code protograph will
lead to inevitable short cycles, and show that these short cycles can be
eliminated using a "squashing" procedure that results in a high-girth QC LDPC
code, although not a hierarchical one. We illustrate our approach with designed
examples of girth-10 QC LDPC codes obtained from protographs of one-sided
spatially-coupled codes.Comment: Submitted to IEEE Transactions on Information THeor
Deriving Good LDPC Convolutional Codes from LDPC Block Codes
Low-density parity-check (LDPC) convolutional codes are capable of achieving
excellent performance with low encoding and decoding complexity. In this paper
we discuss several graph-cover-based methods for deriving families of
time-invariant and time-varying LDPC convolutional codes from LDPC block codes
and show how earlier proposed LDPC convolutional code constructions can be
presented within this framework. Some of the constructed convolutional codes
significantly outperform the underlying LDPC block codes. We investigate some
possible reasons for this "convolutional gain," and we also discuss the ---
mostly moderate --- decoder cost increase that is incurred by going from LDPC
block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010;
revised August 2010, revised November 2010 (essentially final version).
(Besides many small changes, the first and second revised versions contain
corrected entries in Tables I and II.
Codes on Graphs and More
Modern communication systems strive to achieve reliable and efficient information transmission and storage with affordable complexity. Hence, efficient low-complexity channel codes providing low probabilities for erroneous receptions are needed. Interpreting codes as graphs and graphs as codes opens new perspectives for constructing such channel codes. Low-density parity-check (LDPC) codes are one of the most recent examples of codes defined on graphs, providing a better bit error probability than other block codes, given the same decoding complexity. After an introduction to coding theory, different graphical representations for channel codes are reviewed. Based on ideas from graph theory, new algorithms are introduced to iteratively search for LDPC block codes with large girth and to determine their minimum distance. In particular, new LDPC block codes of different rates and with girth up to 24 are presented. Woven convolutional codes are introduced as a generalization of graph-based codes and an asymptotic bound on their free distance, namely, the Costello lower bound, is proven. Moreover, promising examples of woven convolutional codes are given, including a rate 5/20 code with overall constraint length 67 and free distance 120. The remaining part of this dissertation focuses on basic properties of convolutional codes. First, a recurrent equation to determine a closed form expression of the exact decoding bit error probability for convolutional codes is presented. The obtained closed form expression is evaluated for various realizations of encoders, including rate 1/2 and 2/3 encoders, of as many as 16 states. Moreover, MacWilliams-type identities are revisited and a recursion for sequences of spectra of truncated as well as tailbitten convolutional codes and their duals is derived. Finally, the dissertation is concluded with exhaustive searches for convolutional codes of various rates with either optimum free distance or optimum distance profile, extending previously published results
Construction of LDPC Codes Using Randomly Permutated Copies of Parity Check Matrix
Low-density parity-check codes (LDPC) have been shown to have good error correcting performance, putting in mind the Shannon's limit approaching capability. This enables an efficient and reliable communication. However, the construction method of LDPC code can vary over a wide range of parameters such as rate, girth and length. There is a need to develop methods of constructing codes over a wide range of rates and lengths with good performance. This research studies the construction of LDPC codes in randomized and structured form. The contribution of this thesis is introducing a method called "Randomly permutated copies of parity check matrix" that uses a base parity check matrix designed by a random or structured construction method such as Gallager or QC-LDPC codes respectively to get codes with multiple lengths and same rate of the base matrix. This is done by using a seed matrix with row and column weights of one, distributed randomly and can be addressed by a number in the base matrix. This method reduces the memory space needed for storing large parity check matrices, and also reduces the probability of failing to construct a parity matrix by random approaches. Numerical results show that the proposed construction performs similarly to random codes with the same length and rate as in Gallager's and Mackay's codes. It also increases the girth average of the Tanner graph and reduces the number of 4 cycles in the resulted matrix if exists in a base graph