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Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attain many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both…
Polar encoding, described by Arikan in IEEE Transactions on Information Theory, Vol. 55, No. 7, July 2009, was a milestone for telecommunications. A Polar code distributes information among high and low-capacity channels, showing the…
A lower bound on minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of generalized cosets of the codes generated by bottom rows of the polarizing matrix. Moreover, a construction of…
The min-sum approximation is widely used in the decoding of polar codes. Although it is a numerical approximation, hardly any penalties are incurred in practice. We give a theoretical justification for this. We consider the common case of a…
Polar codes are a recent family of error-correcting codes with a number of desirable characteristics. Their disruptive nature is illustrated by their rapid adoption in the $5^{th}$-generation mobile-communication standard, where they are…
This paper presents a refined analysis of the block error rate (BLER) of polar codes over symmetric binary-input discrete memoryless channels under successive cancellation (SC) and successive cancellation list (SCL) decoding. A novel…
Polar code lattices are formed from binary polar codes using Construction D. In this paper, we propose a design technique for finite-dimension polar code lattices. The dimension $n$ and target probability of decoding error are parameters…
Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated…
A packing lemma is proved using a setting where the channel is a binary-input discrete memoryless channel $(\mathcal{X},w(y|x),\mathcal{Y})$, the code is selected at random subject to parity-check constraints, and the decoder is a joint…
Polar codes provably achieve the capacity of a wide array of channels under successive decoding. This assumes infinite precision arithmetic. Given the successive nature of the decoding algorithm, one might worry about the sensitivity of the…
We introduce the design of a set of code sequences $ \{ {\mathscr C}_{n}^{(m)} : n\geq 1, m \geq 1 \}$, with memory order $m$ and code-length $N=O(\phi^n)$, where $ \phi \in (1,2]$ is the largest real root of the polynomial equation…
Arikan's recursive code construction is designed to polarize a collection of memoryless channels into a set of good and a set of bad channels, and it can be efficiently decoded using successive cancellation. It was recently shown that the…
Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_e$ be the block error probability under successive cancellation decoding. In this paper, we develop new…
We propose a new polar code construction framework (i.e., selecting the frozen bit positions) for the additive white Gaussian noise (AWGN) channel, tailored to a given decoding algorithm, rather than based on the (not necessarily optimal)…
The structure of a linear block code is pivotal in defining fundamental properties, particularly weight distribution, and code design. In this study, we characterize the Type II structure of polar codewords with weights less than twice the…
The polar receiver architecture is a receiver design that captures the envelope and phase information of the signal rather than its in-phase and quadrature components. Several studies have demonstrated the robustness of polar receivers to…
Low check weight is practically crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we explore the theory of weight-constrained stabilizer…
When a neural network (NN) is used to decode a polar code, its training complexity scales exponentially as the code block size (or to be precise, as a number of message bits) increases. Therefore, existing solutions that use a neural…
This paper investigates properties of polar codes that can be potentially useful in real-world applications. We start with analyzing the performance of finite-length polar codes over the binary erasure channel (BEC), while assuming belief…
A generalization of the polar coding scheme called mixed-kernels is introduced. This generalization exploits several homogeneous kernels over alphabets of different sizes. An asymptotic analysis of the proposed scheme shows that its…