arXiv subjects

Ido Tal

Publications and source records attributed to Ido Tal.

26 records · Page 2Linked to original sources

arXiv2013

How to Construct Polar Codes

A method for efficiently constructing polar codes is presented and analyzed. Although polar codes are explicitly defined, straightforward construction is intractable since the resulting polar bit-channels have an output alphabet that grows exponentially with he code length. Thus the core problem that needs to be solved is that of faithfully approximating a bit-channel with an intractably large alphabet by another channel having a manageable alphabet size. We devise two approximation methods which "sandwich" the original bit-channel between a degraded and an upgraded version thereof. Both approximations can be efficiently computed, and turn out to be extremely close in practice. We also provide theoretical analysis of our construction algorithms, proving that for any fixed $ε> 0$ and all sufficiently large code lengths $n$, polar codes whose rate is within $ε$ of channel capacity can be constructed in time and space that are both linear in $n$.

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arXiv2012

On the Construction of Polar Codes

We consider the problem of efficiently constructing polar codes over binary memoryless symmetric (BMS) channels. The complexity of designing polar codes via an exact evaluation of the polarized channels to find which ones are "good" appears to be exponential in the block length. In \cite{TV11}, Tal and Vardy show that if instead the evaluation if performed approximately, the construction has only linear complexity. In this paper, we follow this approach and present a framework where the algorithms of \cite{TV11} and new related algorithms can be analyzed for complexity and accuracy. We provide numerical and analytical results on the efficiency of such algorithms, in particular we show that one can find all the "good" channels (except a vanishing fraction) with almost linear complexity in block-length (except a polylogarithmic factor).

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arXiv2012

List Decoding of Polar Codes

We describe a successive-cancellation \emph{list} decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arıkan. In the proposed list decoder, up to $L$ decoding paths are considered concurrently at each decoding stage. Then, a single codeword is selected from the list as output. If the most likely codeword is selected, simulation results show that the resulting performance is very close to that of a maximum-likelihood decoder, even for moderate values of $L$. Alternatively, if a "genie" is allowed to pick the codeword from the list, the results are comparable to the current state of the art LDPC codes. Luckily, implementing such a helpful genie is easy. Our list decoder doubles the number of decoding paths at each decoding step, and then uses a pruning procedure to discard all but the $L$ "best" paths. %In order to implement this algorithm, we introduce a natural pruning criterion that can be easily evaluated. Nevertheless, a straightforward implementation still requires $Ω(L \cdot n^2)$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. We utilize the structure of polar codes to overcome this problem. Specifically, we devise an efficient, numerically stable, implementation taking only $O(L \cdot n \log n)$ time and $O(L \cdot n)$ space.

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arXiv2011

Hardware Implementation of Successive Cancellation Decoders for Polar Codes

The recently-discovered polar codes are seen as a major breakthrough in coding theory; they provably achieve the theoretical capacity of discrete memoryless channels using the low complexity successive cancellation (SC) decoding algorithm. Motivated by recent developments in polar coding theory, we propose a family of efficient hardware implementations for SC polar decoders. We show that such decoders can be implemented with O(n) processing elements, O(n) memory elements, and can provide a constant throughput for a given target clock frequency. Furthermore, we show that SC decoding can be implemented in the logarithm domain, thereby eliminating costly multiplication and division operations and reducing the complexity of each processing element greatly. We also present a detailed architecture for an SC decoder and provide logic synthesis results confirming the linear growth in complexity of the decoder as the code length increases.

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arXiv2010

Hardware architectures for Successive Cancellation Decoding of Polar Codes

The recently-discovered polar codes are widely seen as a major breakthrough in coding theory. These codes achieve the capacity of many important channels under successive cancellation decoding. Motivated by the rapid progress in the theory of polar codes, we propose a family of architectures for efficient hardware implementation of successive cancellation decoders. We show that such decoders can be implemented with O(n) processing elements and O(n) memory elements, while providing constant throughput. We also propose a technique for overlapping the decoding of several consecutive codewords, thereby achieving a significant speed-up factor. We furthermore show that successive cancellation decoding can be implemented in the logarithmic domain, thereby eliminating the multiplication and division operations and greatly reducing the complexity of each processing element.

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arXiv2009

Concave Programming Upper Bounds on the Capacity of 2-D Constraints

The capacity of 1-D constraints is given by the entropy of a corresponding stationary maxentropic Markov chain. Namely, the entropy is maximized over a set of probability distributions, which is defined by some linear requirements. In this paper, certain aspects of this characterization are extended to 2-D constraints. The result is a method for calculating an upper bound on the capacity of 2-D constraints. The key steps are: The maxentropic stationary probability distribution on square configurations is considered. A set of linear equalities and inequalities is derived from this stationarity. The result is a concave program, which can be easily solved numerically. Our method improves upon previous upper bounds for the capacity of the 2-D ``no independent bits'' constraint, as well as certain 2-D RLL constraints.

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arXiv2009

Bounds on the Rate of 2-D Bit-Stuffing Encoders

A method for bounding the rate of bit-stuffing encoders for 2-D constraints is presented. Instead of considering the original encoder, we consider a related one which is quasi-stationary. We use the quasi-stationary property in order to formulate linear requirements that must hold on the probabilities of the constrained arrays that are generated by the encoder. These requirements are used as part of a linear program. The minimum and maximum of the linear program bound the rate of the encoder from below and from above, respectively. A lower bound on the rate of an encoder is also a lower bound on the capacity of the corresponding constraint. For some constraints, our results lead to tighter lower bounds than what was previously known.

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arXiv2008

On row-by-row coding for 2-D constraints

A constant-rate encoder--decoder pair is presented for a fairly large family of two-dimensional (2-D) constraints. Encoding and decoding is done in a row-by-row manner, and is sliding-block decodable. Essentially, the 2-D constraint is turned into a set of independent and relatively simple one-dimensional (1-D) constraints; this is done by dividing the array into fixed-width vertical strips. Each row in the strip is seen as a symbol, and a graph presentation of the respective 1-D constraint is constructed. The maxentropic stationary Markov chain on this graph is next considered: a perturbed version of the corresponding probability distribution on the edges of the graph is used in order to build an encoder which operates in parallel on the strips. This perturbation is found by means of a network flow, with upper and lower bounds on the flow through the edges. A key part of the encoder is an enumerative coder for constant-weight binary words. A fast realization of this coder is shown, using floating-point arithmetic.

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