794 research outputs found
Source and Channel Polarization over Finite Fields and Reed-Solomon Matrices
Polarization phenomenon over any finite field with size
being a power of a prime is considered. This problem is a generalization of the
original proposal of channel polarization by Arikan for the binary field, as
well as its extension to a prime field by Sasoglu, Telatar, and Arikan. In this
paper, a necessary and sufficient condition of a matrix over a finite field
is shown under which any source and channel are polarized.
Furthermore, the result of the speed of polarization for the binary alphabet
obtained by Arikan and Telatar is generalized to arbitrary finite field. It is
also shown that the asymptotic error probability of polar codes is improved by
using the Reed-Solomon matrix, which can be regarded as a natural
generalization of the binary matrix used in the original proposal
by Arikan.Comment: 17 pages, 3 figures, accepted for publication in the IEEE
Transactions on Information Theor
Polar Codes: Characterization of Exponent, Bounds, and Constructions
Polar codes were recently introduced by Ar\i kan. They achieve the capacity
of arbitrary symmetric binary-input discrete memoryless channels under a low
complexity successive cancellation decoding strategy. The original polar code
construction is closely related to the recursive construction of Reed-Muller
codes and is based on the matrix \bigl[ 1 &0 1& 1 \bigr]. It was
shown by Ar\i kan and Telatar that this construction achieves an error exponent
of , i.e., that for sufficiently large blocklengths the error
probability decays exponentially in the square root of the length. It was
already mentioned by Ar\i kan that in principle larger matrices can be used to
construct polar codes. A fundamental question then is to see whether there
exist matrices with exponent exceeding . We first show that any matrix none of whose column permutations is upper triangular
polarizes symmetric channels. We then characterize the exponent of a given
square matrix and derive upper and lower bounds on achievable exponents. Using
these bounds we show that there are no matrices of size less than 15 with
exponents exceeding . Further, we give a general construction based on
BCH codes which for large achieves exponents arbitrarily close to 1 and
which exceeds for size 16.Comment: Submitted to IEEE Transactions on Information Theory, minor update
Concatenation of convolutional and block codes Final report
Comparison of concatenated and sequential decoding systems and convolutional code structural propertie
Quantum coding with finite resources
The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. Here we illustrate that this asymptotic characterization is insufficient in practical scenarios where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. In practical settings, we should instead focus on the optimal trade-off between three parameters: the rate of the code, the size of the quantum devices at the encoder and decoder, and the fidelity of the transmission. We find approximate and exact characterizations of this trade-off for various channels of interest, including dephasing, depolarizing and erasure channels. In each case, the trade-off is parameterized by the capacity and a second channel parameter, the quantum channel dispersion. In the process, we develop several bounds that are valid for general quantum channels and can be computed for small instances
Saddle Point in the Minimax Converse for Channel Coding
A minimax metaconverse has recently been proposed as a simultaneous generalization of a number of classical results and a tool for the nonasymptotic analysis. In this paper, it is shown that the order of optimizing the input and output distributions can be interchanged without affecting the bound. In the course of the proof, a number of auxiliary results of separate interest are obtained. In particular, it is shown that the optimization problem is convex and can be solved in many cases by the symmetry considerations. As a consequence, it is demonstrated that in the latter cases, the (multiletter) input distribution in information-spectrum (Verdú-Han) converse bound can be taken to be a (memoryless) product of single-letter ones. A tight converse for the binary erasure channel is rederived by computing the optimal (nonproduct) output distribution. For discrete memoryless channels, a conjecture of Poor and Verdú regarding the tightness of the information spectrum bound on the error exponents is resolved in the negative. Concept of the channel symmetry group is established and relations with the definitions of symmetry by Gallager and Dobrushin are investigated.National Science Foundation (U.S.) (Center for Science of Information, under Grant CCF-0939370
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