Eren Sasoglu

Polarization for arbitrary discrete memoryless channels

Channel polarization, originally proposed for binary-input channels, is generalized to arbitrary discrete memoryless 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 memoryless 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.

Polar Codes for the Two-User Multiple-Access Channel

Arikans polar coding method is extended to two-user multiple-access channels. It is shown that if the two users of the channel use Arikans construction, the resulting channels will polarize to one of five possible extremals, on each of which uncoded transmission is optimal. The sum rate achieved by this coding technique is the one that corresponds to uniform input distributions. The encoding and decoding complexities and the error performance of these codes are as in the single-user case: O(nlogn) for encoding and decoding, and o(2-n1/2-ε) for the block error probability, where n is the blocklength.

On the capacity region for index coding

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).

Polar codes: Characterization of exponent, bounds, and constructions

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.

A new polar coding scheme for strong security on wiretap channels

The problem of achieving the secrecy capacity of wiretap channels explicitly and with low complexity has been open since the work of Wyner in 1975. Recently, Mahdavifar and Vardy presented a solution to this problem, based on polar codes, for the class of symmetric and degraded wiretap channels. Their polar coding scheme achieves both security and reliability under the weak security criterion, but does not guarantee reliability under the strong security criterion. The main difficulty in providing both strong security and reliability using polar codes is the existence of a small number of bit-channels that are both unreliable and unsecure. In this paper, a multi-block polar coding scheme that resolves this difficulty is presented. It is shown that this coding scheme achieves the secrecy capacity of symmetric degraded wiretap channels while guaranteeing both reliability and strong security.

Polar codes for discrete alphabets

An open problem in polarization theory is whether all memoryless channels and sources with composite (that is, non-prime) alphabet sizes can be polarized with deterministic, Arıkan-like methods. This paper answers the question in the affirmative by giving a method to polarize all discrete memoryless channels and sources. The method yields codes that retain the low encoding and decoding complexity of binary polar codes.

Polar coding for interference networks

A polar coding scheme for interference networks is introduced. The scheme builds on Arikans monotone chain rules for multiple access channels and a method by Hassani and Urbanke to “align” two incompatible polarization processes. It achieves the Han-Kobayashi inner bound for two-user interference channels and generalizes to interference networks.

Sliding-Window Superposition Coding for Interference Networks

Superposition coding with successive cancellation decoding for interference channels is investigated as a low-complexity alternative to the rate-optimal simultaneous decoding. It is shown that regardless of the number of superposition layers and the code distribution of each layer, the standard rate-splitting scheme by Grant, Rimoldi, Urbanke, and Whiting for multiple access channels fails to achieve the simultaneous decoding inner bound on the capacity region for interference channels. A new coding scheme is proposed that uses coding over multiple blocks and sliding-window decoding. With at most two superposition layers, this scheme achieves the simultaneous decoding inner bound for any two-user-pair interference channels without using high-complexity simultaneous multiuser sequence detection. The proposed coding scheme can be also extended to achieve the performance of simultaneous decoding for general interference networks, including the Han-Kobayashi inner bound.

A comparison of superposition coding schemes

There are two variants of superposition coding schemes. Covers original superposition coding scheme has code clouds of identical shape, while Bergmanss superposition coding scheme has code clouds of independently generated shapes. These two schemes yield identical achievable rate regions in several scenarios, such as the capacity region for degraded broadcast channels. This paper shows that under optimal decoding, these two superposition coding schemes can result in different rate regions. In particular, it is shown that for the two-receiver broadcast channel, Covers scheme achieves a larger rate region than Bergmanss scheme in general.

An entropy inequality for q-ary random variables and its application to channel polarization

It is shown that given two copies of a q-ary input channel W, where q is prime, it is possible to create two channels W⁻ and W⁺ whose symmetric capacities satisfy I(W⁻) ≤ I(W) ≤ I(W⁺), where the inequalities are strict except in trivial cases. This leads to a simple proof of channel polarization in the q-ary case.