233 research outputs found
Spatially-Coupled QDLPC Codes
Spatially-coupled (SC) codes is a class of convolutional LDPC codes that has
been well investigated in classical coding theory thanks to their high
performance and compatibility with low-latency decoders. We describe toric
codes as quantum counterparts of classical two-dimensional spatially-coupled
(2D-SC) codes, and introduce spatially-coupled quantum LDPC (SC-QLDPC) codes as
a generalization. We use the convolutional structure to represent the parity
check matrix of a 2D-SC code as a polynomial in two indeterminates, and derive
an algebraic condition that is both necessary and sufficient for a 2D-SC code
to be a stabilizer code. This algebraic framework facilitates the construction
of new code families. While not the focus of this paper, we note that small
memory facilitates physical connectivity of qubits, and it enables local
encoding and low-latency windowed decoding. In this paper, we use the algebraic
framework to optimize short cycles in the Tanner graph of 2D-SC HGP codes that
arise from short cycles in either component code. While prior work focuses on
QLDPC codes with rate less than 1/10, we construct 2D-SC HGP codes with small
memory, higher rates (about 1/3), and superior thresholds.Comment: 25 pages, 7 figure
Variations of the McEliece Cryptosystem
Two variations of the McEliece cryptosystem are presented. The first one is
based on a relaxation of the column permutation in the classical McEliece
scrambling process. This is done in such a way that the Hamming weight of the
error, added in the encryption process, can be controlled so that efficient
decryption remains possible. The second variation is based on the use of
spatially coupled moderate-density parity-check codes as secret codes. These
codes are known for their excellent error-correction performance and allow for
a relatively low key size in the cryptosystem. For both variants the security
with respect to known attacks is discussed
Asymptotic Analysis on Spatial Coupling Coding for Two-Way Relay Channels
Compute-and-forward relaying is effective to increase bandwidth efficiency of
wireless two-way relay channels. In a compute-and-forward scheme, a relay tries
to decode a linear combination composed of transmitted messages from other
terminals or relays. Design for error correcting codes and its decoding
algorithms suitable for compute-and-forward relaying schemes are still
important issue to be studied. In this paper, we will present an asymptotic
performance analysis on LDPC codes over two-way relay channels based on density
evolution (DE). Because of the asymmetric nature of the channel, we employ the
population dynamics DE combined with DE formulas for asymmetric channels to
obtain BP thresholds. In addition, we also evaluate the asymptotic performance
of spatially coupled LDPC codes for two-way relay channels. The results
indicate that the spatial coupling codes yield improvements in the BP threshold
compared with corresponding uncoupled codes for two-way relay channels.Comment: 5 page
Non-Binary LDPC Codes with Large Alphabet Size
We study LDPC codes for the channel with input and
output . The aim of this paper is to evaluate
decoding performance of -ary non-binary LDPC codes for large . We give
density evolution and decoding performance evaluation for regular non-binary
LDPC codes and spatially-coupled (SC) codes. We show the regular codes do not
achieve the capacity of the channel while SC codes do
The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing
and protecting fragile qubits against the undesirable effects of quantum
decoherence. Similar to classical codes, hashing bound approaching QECCs may be
designed by exploiting a concatenated code structure, which invokes iterative
decoding. Therefore, in this paper we provide an extensive step-by-step
tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided
concatenated quantum codes based on the underlying quantum-to-classical
isomorphism. These design lessons are then exemplified in the context of our
proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the
outer component of a concatenated quantum code. The proposed QIRCC can be
dynamically adapted to match any given inner code using EXIT charts, hence
achieving a performance close to the hashing bound. It is demonstrated that our
QIRCC-based optimized design is capable of operating within 0.4 dB of the noise
limit
New Codes on Graphs Constructed by Connecting Spatially Coupled Chains
A novel code construction based on spatially coupled low-density parity-check
(SC-LDPC) codes is presented. The proposed code ensembles are described by
protographs, comprised of several protograph-based chains characterizing
individual SC-LDPC codes. We demonstrate that code ensembles obtained by
connecting appropriately chosen SC-LDPC code chains at specific points have
improved iterative decoding thresholds compared to those of single SC-LDPC
coupled chains. In addition, it is shown that the improved decoding properties
of the connected ensembles result in reduced decoding complexity required to
achieve a specific bit error probability. The constructed ensembles are also
asymptotically good, in the sense that the minimum distance grows linearly with
the block length. Finally, we show that the improved asymptotic properties of
the connected chain ensembles also translate into improved finite length
performance.Comment: Submitted to IEEE Transactions on Information Theor
Spatially Coupled LDPC Codes Constructed from Protographs
In this paper, we construct protograph-based spatially coupled low-density
parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or
uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,
we obtain a flexible family of code ensembles with varying rates and frame
lengths that can share the same encoding and decoding architecture for
arbitrary L. We demonstrate that the resulting codes combine the best features
of optimized irregular and regular codes in one design: capacity approaching
iterative belief propagation (BP) decoding thresholds and linear growth of
minimum distance with block length. In particular, we show that, for
sufficiently large L, the BP thresholds on both the binary erasure channel
(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)
saturate to a particular value significantly better than the BP decoding
threshold and numerically indistinguishable from the optimal maximum
a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all
variable nodes in the coupled chain have degree greater than two,
asymptotically the error probability converges at least doubly exponentially
with decoding iterations and we obtain sequences of asymptotically good LDPC
codes with fast convergence rates and BP thresholds close to the Shannon limit.
Further, the gap to capacity decreases as the density of the graph increases,
opening up a new way to construct capacity achieving codes on memoryless
binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor
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