32 research outputs found
Decoding of Convolutional Codes over the Erasure Channel
In this paper we study the decoding capabilities of convolutional codes over
the erasure channel. Of special interest will be maximum distance profile (MDP)
convolutional codes. These are codes which have a maximum possible column
distance increase. We show how this strong minimum distance condition of MDP
convolutional codes help us to solve error situations that maximum distance
separable (MDS) block codes fail to solve. Towards this goal, we define two
subclasses of MDP codes: reverse-MDP convolutional codes and complete-MDP
convolutional codes. Reverse-MDP codes have the capability to recover a maximum
number of erasures using an algorithm which runs backward in time. Complete-MDP
convolutional codes are both MDP and reverse-MDP codes. They are capable to
recover the state of the decoder under the mildest condition. We show that
complete-MDP convolutional codes perform in certain sense better than MDS block
codes of the same rate over the erasure channel.Comment: 18 pages, 3 figures, to appear on IEEE Transactions on Information
Theor
Decoding of MDP Convolutional Codes over the Erasure Channel
This paper studies the decoding capabilities of maximum distance profile
(MDP) convolutional codes over the erasure channel and compares them with the
decoding capabilities of MDS block codes over the same channel. The erasure
channel involving large alphabets is an important practical channel model when
studying packet transmissions over a network, e.g, the Internet
Absdet-Pseudo-Codewords and Perm-Pseudo-Codewords: Definitions and Properties
The linear-programming decoding performance of a binary linear code crucially
depends on the structure of the fundamental cone of the parity-check matrix
that describes the code. Towards a better understanding of fundamental cones
and the vectors therein, we introduce the notion of absdet-pseudo-codewords and
perm-pseudo-codewords: we give the definitions, we discuss some simple
examples, and we list some of their properties
On the Minimal Pseudo-Codewords of Codes from Finite Geometries
In order to understand the performance of a code under maximum-likelihood
(ML) decoding, it is crucial to know the minimal codewords. In the context of
linear programming (LP) decoding, it turns out to be necessary to know the
minimal pseudo-codewords. This paper studies the minimal codewords and minimal
pseudo-codewords of some families of codes derived from projective and
Euclidean planes. Although our numerical results are only for codes of very
modest length, they suggest that these code families exhibit an interesting
property. Namely, all minimal pseudo-codewords that are not multiples of a
minimal codeword have an AWGNC pseudo-weight that is strictly larger than the
minimum Hamming weight of the code. This observation has positive consequences
not only for LP decoding but also for iterative decoding.Comment: To appear in Proc. 2005 IEEE International Symposium on Information
Theory, Adelaide, Australia, September 4-9, 200
Quasi-Cyclic Asymptotically Regular LDPC Codes
Families of "asymptotically regular" LDPC block code ensembles can be formed
by terminating (J,K)-regular protograph-based LDPC convolutional codes. By
varying the termination length, we obtain a large selection of LDPC block code
ensembles with varying code rates, minimum distance that grows linearly with
block length, and capacity approaching iterative decoding thresholds, despite
the fact that the terminated ensembles are almost regular. In this paper, we
investigate the properties of the quasi-cyclic (QC) members of such an
ensemble. We show that an upper bound on the minimum Hamming distance of
members of the QC sub-ensemble can be improved by careful choice of the
component protographs used in the code construction. Further, we show that the
upper bound on the minimum distance can be improved by using arrays of
circulants in a graph cover of the protograph.Comment: To be presented at the 2010 IEEE Information Theory Workshop, Dublin,
Irelan