arXiv subjects
Publications and source records attributed to Erdal Arikan.
Performance and complexity of sequential decoding of polarization-adjusted convolutional (PAC) codes is studied. In particular, a performance and computational complexity comparison of PAC codes with 5G polar codes and convolutional codes is given. A method for bounding the complexity of sequential decoding of PAC codes is proposed.
We give a unified treatment of some inequalities that are used in the proofs of channel polarization theorems involving a binary-input discrete memoryless channel.
Three areas of ongoing research in channel coding are surveyed, and recent developments are presented in each area: spatially coupled Low-Density Parity-Check (LDPC) codes, non-binary LDPC codes, and polar coding.
The notion of source polarization is introduced and investigated. This complements the earlier work on channel polarization. An application to Slepian-Wolf coding is also considered. The paper is restricted to the case of binary alphabets. Extension of results to non-binary alphabets is discussed briefly.
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.
A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete memoryless channel (B-DMC) $W$. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of $N$ independent copies of a given B-DMC $W$, a second set of $N$ binary-input channels $\{W_N^{(i)}:1\le i\le N\}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ is near 1 approaches $I(W)$ and the fraction for which $I(W_N^{(i)})$ is near 0 approaches $1-I(W)$. The polarized channels $\{W_N^{(i)}\}$ are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC $W$ with $I(W)>0$ and any target rate $R < I(W)$, there exists a sequence of polar codes $\{{\mathscr C}_n;n\ge 1\}$ such that ${\mathscr C}_n$ has block-length $N=2^n$, rate $\ge R$, and probability of block error under successive cancellation decoding bounded as $P_{e}(N,R) \le \bigoh(N^{-\frac14})$ independently of the code rate. This performance is achievable by encoders and decoders with complexity $O(N\log N)$ for each.
It is shown that for any binary-input discrete memoryless channel $W$ with symmetric capacity $I(W)$ and any rate $R <I(W)$, the probability of block decoding error for polar coding under successive cancellation decoding satisfies $P_e \le 2^{-N^β}$ for any $β<\frac12$ when the block-length $N$ is large enough.
The cutoff rate $R_0(W)$ of a discrete memoryless channel (DMC) $W$ is often used as a figure of merit, alongside the channel capacity $C(W)$. Given a channel $W$ consisting of two possibly correlated subchannels $W_1$, $W_2$, the capacity function always satisfies $C(W_1)+C(W_2) \le C(W)$, while there are examples for which $R_0(W_1)+R_0(W_2) > R_0(W)$. This fact that cutoff rate can be ``created'' by channel splitting was noticed by Massey in his study of an optical modulation system modeled as a $M$'ary erasure channel. This paper demonstrates that similar gains in cutoff rate can be achieved for general DMC's by methods of channel combining and splitting. Relation of the proposed method to Pinsker's early work on cutoff rate improvement and to Imai-Hirakawa multi-level coding are also discussed.
Grover discovered a quantum algorithm for identifying a target element in an unstructured search universe of N items in approximately square-root of N queries to a quantum oracle, thus achieving a square-root speed-up over classical algorithms. We present an information-theoretic analysis of Grover's algorithm and show that the square-root speed-up is the best attainable result using Grover's oracle.