5,861 research outputs found
Tensor Networks and Quantum Error Correction
We establish several relations between quantum error correction (QEC) and
tensor network (TN) methods of quantum many-body physics. We exhibit
correspondences between well-known families of QEC codes and TNs, and
demonstrate a formal equivalence between decoding a QEC code and contracting a
TN. We build on this equivalence to propose a new family of quantum codes and
decoding algorithms that generalize and improve upon quantum polar codes and
successive cancellation decoding in a natural way.Comment: Accepted in Phys. Rev. Lett. 8 pages, 9 figure
Construction of Capacity-Achieving Lattice Codes: Polar Lattices
In this paper, we propose a new class of lattices constructed from polar
codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR)
of the additive white Gaussian-noise (AWGN) channel. Our construction follows
the multilevel approach of Forney \textit{et al.}, where we construct a
capacity-achieving polar code on each level. The component polar codes are
shown to be naturally nested, thereby fulfilling the requirement of the
multilevel lattice construction. We prove that polar lattices are
\emph{AWGN-good}. Furthermore, using the technique of source polarization, we
propose discrete Gaussian shaping over the polar lattice to satisfy the power
constraint. Both the construction and shaping are explicit, and the overall
complexity of encoding and decoding is for any fixed target error
probability.Comment: full version of the paper to appear in IEEE Trans. Communication
Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes
In this work, we present hardware and software implementations of flexible
polar systematic encoders and decoders. The proposed implementations operate on
polar codes of any length less than a maximum and of any rate. We describe the
low-complexity, highly parallel, and flexible systematic-encoding algorithm
that we use and prove its correctness. Our hardware implementation results show
that the overhead of adding code rate and length flexibility is little, and the
impact on operation latency minor compared to code-specific versions. Finally,
the flexible software encoder and decoder implementations are also shown to be
able to maintain high throughput and low latency.Comment: Submitted to IEEE Transactions on Communications, 201
Sparse Regression Codes for Multi-terminal Source and Channel Coding
We study a new class of codes for Gaussian multi-terminal source and channel
coding. These codes are designed using the statistical framework of
high-dimensional linear regression and are called Sparse Superposition or
Sparse Regression codes. Codewords are linear combinations of subsets of
columns of a design matrix. These codes were recently introduced by Barron and
Joseph and shown to achieve the channel capacity of AWGN channels with
computationally feasible decoding. They have also recently been shown to
achieve the optimal rate-distortion function for Gaussian sources. In this
paper, we demonstrate how to implement random binning and superposition coding
using sparse regression codes. In particular, with minimum-distance
encoding/decoding it is shown that sparse regression codes attain the optimal
information-theoretic limits for a variety of multi-terminal source and channel
coding problems.Comment: 9 pages, appeared in the Proceedings of the 50th Annual Allerton
Conference on Communication, Control, and Computing - 201
Polar Codes for the m-User MAC
In this paper, polar codes for the -user multiple access channel (MAC)
with binary inputs are constructed. It is shown that Ar{\i}kan's polarization
technique applied individually to each user transforms independent uses of a
-user binary input MAC into successive uses of extremal MACs. This
transformation has a number of desirable properties: (i) the `uniform sum rate'
of the original MAC is preserved, (ii) the extremal MACs have uniform rate
regions that are not only polymatroids but matroids and thus (iii) their
uniform sum rate can be reached by each user transmitting either uncoded or
fixed bits; in this sense they are easy to communicate over. A polar code can
then be constructed with an encoding and decoding complexity of
(where is the block length), a block error probability of o(\exp(- n^{1/2
- \e})), and capable of achieving the uniform sum rate of any binary input MAC
with arbitrary many users. An application of this polar code construction to
communicating on the AWGN channel is also discussed
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