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Balanced codes
Balanced codes, in which each codeword contains equally many 1's and 0's, are useful in such applications as in optical transmission and optical recording. When balanced codes are used, the same number of 1's and 0's pass through the channel after the transmission of every word, so the channel is in a dc-null state. Optical channels require this property because they employ AC-coupled devices. Line codes, in which codewords may not be balanced, are also used as dc-free codes in such channels. In this thesis we present the research that leads to the following results: 1- Balanced codes These have higher information rate than existing codes yet maintain similar encoding and decoding complexities. 2- Error-correcting balanced codes In many cases, these give higher information rates and more efficient encoding and decoding algorithms than the best-known equivalent codes. 3- DC-Free coset codes A new technique to design dc-free coset codes was developed. These codes have better properties than existing ones. 4- Generalization of balanced codes -- Balanced codes are generalized in three ways among which the first is the most significant: a) Balanced codes with low dc level These codes are designed based on the combined techniques used in (1) and (3) above. A lower dc-level and higher transitions density is achieved at the cost of one extra check bit. These codes are much more attractive, to optical transmission, than the bare-bone balanced codes. b) Non-Binary Balanced Codes Balanced codes over a non-binary alphabet. c) Semi-Balanced Codes -- Codes in which the number of 1's and 0's in every code word differs by at most a certain value. 5- t-EC/AUED coset codes These are t error correcting/all unidirectional error detecting codes. Again the technique in (3) above is used to design t-EC/AUED coset codes. These codes obtain higher information rate than the best-known equivalent codes and yet maintain the same encoding/decoding complexity
Density Evolution for Asymmetric Memoryless Channels
Density evolution is one of the most powerful analytical tools for
low-density parity-check (LDPC) codes and graph codes with message passing
decoding algorithms. With channel symmetry as one of its fundamental
assumptions, density evolution (DE) has been widely and successfully applied to
different channels, including binary erasure channels, binary symmetric
channels, binary additive white Gaussian noise channels, etc. This paper
generalizes density evolution for non-symmetric memoryless channels, which in
turn broadens the applications to general memoryless channels, e.g. z-channels,
composite white Gaussian noise channels, etc. The central theorem underpinning
this generalization is the convergence to perfect projection for any fixed size
supporting tree. A new iterative formula of the same complexity is then
presented and the necessary theorems for the performance concentration theorems
are developed. Several properties of the new density evolution method are
explored, including stability results for general asymmetric memoryless
channels. Simulations, code optimizations, and possible new applications
suggested by this new density evolution method are also provided. This result
is also used to prove the typicality of linear LDPC codes among the coset code
ensemble when the minimum check node degree is sufficiently large. It is shown
that the convergence to perfect projection is essential to the belief
propagation algorithm even when only symmetric channels are considered. Hence
the proof of the convergence to perfect projection serves also as a completion
of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor
Product Construction of Affine Codes
Binary matrix codes with restricted row and column weights are a desirable
method of coded modulation for power line communication. In this work, we
construct such matrix codes that are obtained as products of affine codes -
cosets of binary linear codes. Additionally, the constructions have the
property that they are systematic. Subsequently, we generalize our construction
to irregular product of affine codes, where the component codes are affine
codes of different rates.Comment: 13 pages, to appear in SIAM Journal on Discrete Mathematic
On the performance of 1-level LDPC lattices
The low-density parity-check (LDPC) lattices perform very well in high
dimensions under generalized min-sum iterative decoding algorithm. In this work
we focus on 1-level LDPC lattices. We show that these lattices are the same as
lattices constructed based on Construction A and low-density lattice-code
(LDLC) lattices. In spite of having slightly lower coding gain, 1-level regular
LDPC lattices have remarkable performances. The lower complexity nature of the
decoding algorithm for these type of lattices allows us to run it for higher
dimensions easily. Our simulation results show that a 1-level LDPC lattice of
size 10000 can work as close as 1.1 dB at normalized error probability (NEP) of
.This can also be reported as 0.6 dB at symbol error rate (SER) of
with sum-product algorithm.Comment: 1 figure, submitted to IWCIT 201
On Secure Distributed Data Storage Under Repair Dynamics
We address the problem of securing distributed storage systems against
passive eavesdroppers that can observe a limited number of storage nodes. An
important aspect of these systems is node failures over time, which demand a
repair mechanism aimed at maintaining a targeted high level of system
reliability. If an eavesdropper observes a node that is added to the system to
replace a failed node, it will have access to all the data downloaded during
repair, which can potentially compromise the entire information in the system.
We are interested in determining the secrecy capacity of distributed storage
systems under repair dynamics, i.e., the maximum amount of data that can be
securely stored and made available to a legitimate user without revealing any
information to any eavesdropper. We derive a general upper bound on the secrecy
capacity and show that this bound is tight for the bandwidth-limited regime
which is of importance in scenarios such as peer-to-peer distributed storage
systems. We also provide a simple explicit code construction that achieves the
capacity for this regime.Comment: 5 pages, 4 figures, to appear in Proceedings of IEEE ISIT 201
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