Information Theory and Reliable Communication

We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, up to L decoding paths are considered concurrently at each decoding stage. Simulation results show that the resulting performance is very close to that of a maximum-likelihood decoder, even for moderate values of L. Thus it appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes. The specific list-decoding algorithm that achieves this performance 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, straightforward implementation still requires O(L · n2) 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 · n log n) time and O(L · n) space.

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List decoding of polar codes

We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, $L$ decoding paths are considered concurrently at each decoding stage, where $L$ is an integer parameter. At the end of the decoding process, the most likely among the $L$ paths is selected as the single codeword at the decoder output. Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of $L$ . Alternatively, if a genie is allowed to pick the transmitted codeword from the list, the results are comparable with the performance of current state-of-the-art LDPC codes. We show that such a genie can be easily implemented using simple CRC precoding. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit, and then uses a pruning procedure to discard all but the $L$ most likely paths. However, straightforward implementation of this algorithm requires $\Omega (L n^{2})$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. In this paper, we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem: we devise an efficient, numerically stable, implementation of the proposed list decoder that takes only $O(L n \log n)$ time and $O(L n)$ space.

List Decoding of 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 the 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 e > 0 and all sufficiently large code lengths n, polar codes whose rate is within e of channel capacity can be constructed in time and space that are both linear in n.

How to Construct Polar Codes

Polar codes are shown to be instances of both generalized concatenated codes and multilevel codes. It is shown that the performance of a polar code can be improved by representing it as a multilevel code and applying the multistage decoding algorithm with maximum likelihood decoding of outer codes. Additional performance improvement is obtained by replacing polar outer codes with other ones with better error correction performance. In some cases this also results in complexity reduction. It is shown that Gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them.

Efficient Design and Decoding of Polar Codes

Polar codes were recently introduced by Arikan. They achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancellation decoding scheme. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the 2 × 2 matrix [1 0 : 1 1]. It was shown by Arikan Telatar that this construction achieves an error exponent of 1/2, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength. It was already mentioned by Arikan that in principle larger matrices can be used to construct polar codes. In this paper, it is first shown that any l × l matrix none of whose column permutations is upper triangular polarizes binary-input memoryless channels. The exponent of a given square matrix is characterized, upper and lower bounds on achievable exponents are given. Using these bounds it is shown that there are no matrices of size smaller than 15×15 with exponents exceeding 1/2. Further, a general construction based on BCH codes which for large I achieves exponents arbitrarily close to 1 is given. At size 16 × 16, this construction yields an exponent greater than 1/2.

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Polar Codes: Characterization of Exponent, Bounds, and Constructions

Channel polarization, originally proposed for binary-input channels, is generalized to arbitrary discrete mem-oryless channels. Specifically, it is shown that when the input alphabet size is a prime number, a similar construction to that for the binary case leads to polarization. This method can be extended to channels of composite input alphabet sizes by decomposing such channels into a set of channels with prime input alphabet sizes. It is also shown that all discrete memo-ryless channels can be polarized by randomized constructions. The introduction of randomness does not change the order of complexity of polar code construction, encoding, and decoding. A previous result on the error probability behavior of polar codes is also extended to the case of arbitrary discrete memoryless channels. The generalization of polarization to channels with arbitrary finite input alphabet sizes leads to polar-coding methods for approaching the true (as opposed to symmetric) channel capacity of arbitrary channels with discrete or continuous input alphabets.

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Polarization for arbitrary discrete memoryless channels

A new inner bound on the capacity region of the general index coding problem is established. Unlike most existing bounds that are based on graph theoretic or algebraic tools, the bound relies on a random coding scheme and optimal decoding, and has a simple polymatroidal single-letter expression. The utility of the inner bound is demonstrated by examples that include the capacity region for all index coding problems with up to five messages (there are 9846 nonisomorphic ones).

On the capacity region for index coding

In this paper, polar codes for the m-user multiple access channel (MAC) with binary inputs are constructed. It is shown that Arikan's polarization technique applied individually to each user transforms independent uses of an m-user binary input MAC into successive uses of extremal MACs. This transformation has a number of desirable properties: 1) the “uniform sum-rate” of the original MAC is preserved, 2) the extremal MACs have uniform rate regions that are not only polymatroids but matroids, and thus, 3) 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 O(n log n) (where n is the block length), a block error probability of o(exp (- n1/2 - e)), and capable of achieving the uniform sum-rate of any binary input MAC with arbitrary many users. Applications of this polar code construction to channels with a finite field input alphabet and to the additive white Gaussian noise channel are also discussed.

Polar Codes for the Two-User Multiple-Access Channel

Polar codes were recently introduced by Arikan. They achieve the symmetric capacity of arbitrary 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 2 × 2 matrix of the given equation. It was shown by Arikan and Telatar that this construction achieves an error exponent of 1/2, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Arikan 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 1/2. We 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 1/2. Further, we give a general construction based on BCH codes which for large matrix sizes achieves exponents arbitrarily close to 1 and which exceeds 1/2 for size 16.

Eren Sasoglu

Papers

Polar Codes: Characterization of Exponent, Bounds, and Constructions

Polarization for arbitrary discrete memoryless channels

On the capacity region for index coding

Polar Codes for the Two-User Multiple-Access Channel

Polar codes: Characterization of exponent, bounds, and constructions