The polar transformation of a binary erasure channel (BEC) can be exactly approximated by other BECs. Arıkan proposed that polar codes for a BEC can be efficiently constructed by using its useful property. This study proposes a new class of arbitrary input generalized erasure channels, which can be exactly approximated the polar transformation by other same channel models, as with the BEC. One of the main results is the recursive formulas of the polar transformation of the proposed channel. In the study, we evaluate the polar transformation by using the $α$-mutual information. Particularly, when the input alphabet size is a prime power, we examines the following: (i) inequalities for the average of the $α$-mutual information of the proposed channel after the one-step polar transformation, and (ii) the exact proportion of polarizations of the $α$-mutual information of proposed channels in infinite number of polar transformations.
This study examines multilevel channel polarization for a certain class of erasure channels that the input alphabet size is an arbitrary composite number. We derive limiting proportions of partially noiseless channels for such a class. The results of this study are proved by an argument of convergent sequences, inspired by Alsan and Telatar's simple proof of polarization, and without martingale convergence theorems for polarization process.
Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Arıkan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead t…
This study proposes \emph{modular arithmetic erasure channels} (MAECs), a novel class of erasure-like channels with an input alphabet that need not be binary. This class contains the binary erasure channel (BEC) and some other known erasure-like channels as special cases. For MAECs, we provide recursive formulas of Arıkan-like polar transform to simulate channel polarization. In other words, we show that the synthetic channels of MAECs are equivalent to other MAECs. This is a generalization of well-known recursive formulas of the polar transform for BECs. Using our recursive formulas, we also …
We study faulty successive cancellation decoding of polar codes for the binary erasure channel. To this end, we introduce a simple erasure-based fault model and we show that, under this model, polarization does not happen, meaning that fully reliable communication is not possible at any rate. Moreover, we provide numerical results for the frame erasure rate and bit erasure rate and we study an unequal error protection scheme that can significantly improve the performance of the faulty successive cancellation decoder with negligible overhead.
This paper investigates the belief propagation decoding of spatially-coupled MacKay-Neal (SC-MN) codes over erasure channels with memory. We show that SC-MN codes with bounded degree universally achieve the symmetric information rate (SIR) of arbitrary erasure channels with memory. We mean by universality the following sense: the sender does not need to know the whole channel statistics but needs to know only the SIR, while the receiver estimates the transmitted codewords from channel statistics and received words. The proof is based on potential function.
In this paper, we design a new polar slotted ALOHA (PSA) protocol over the slot erasure channels, which uses polar coding to construct the identical slot pattern (SP) assembles within each active user and base station. A theoretical analysis framework for the PSA is provided. First, by using the packet-oriented operation for the overlap packets when they conflict in a slot interval, we introduce the packet-based polarization transform and prove that this transform is independent of the packet's length. Second, guided by the packet-based polarization, an SP assignment (SPA) method with the vari…
In this paper, faulty successive cancellation decoding of polar codes for the binary erasure channel is studied. To this end, a simple erasure-based fault model is introduced to represent errors in the decoder and it is shown that, under this model, polarization does not happen, meaning that fully reliable communication is not possible at any rate. Furthermore, a lower bound on the frame error rate of polar codes under faulty SC decoding is provided, which is then used, along with a well-known upper bound, in order to choose a blocklength that minimizes the erasure probability under faulty dec…
In this paper we study the decoding capabilities of convolutional codes over the erasure channel. Of special interest will be maximum distance profile (MDP) convolutional codes. These are codes which have a maximum possible column distance increase. We show how this strong minimum distance condition of MDP convolutional codes help us to solve error situations that maximum distance separable (MDS) block codes fail to solve. Towards this goal, we define two subclasses of MDP codes: reverse-MDP convolutional codes and complete-MDP convolutional codes. Reverse-MDP codes have the capability to reco…
The two-receiver broadcast packet erasure channel with feedback and memory is studied. Memory is modeled using a finite-state Markov chain representing a channel state. The channel state is unknown at the transmitter, but observations of this hidden Markov chain are available at the transmitter through feedback. Matching outer and inner bounds are derived and the capacity region is determined. The capacity region does not have a single-letter characterization and is, in this sense, uncomputable. Approximations of the capacity region are provided and two optimal coding algorithms are outlined. …
In this paper we describe a network coding scheme for the Broadcast Erasure Channel with multiple unicast stochastic flows, in the case of a single source transmitting packets to $N$ users, where per-slot feedback is fed back to the transmitter in the form of ACK/NACK messages. This scheme performs only binary (XOR) operations and involves a network of queues, along with special rules for coding and moving packets among the queues, that ensure instantaneous decodability. The system under consideration belongs to a class of networks whose stability properties have been analyzed in earlier work,…
In the $q$-ary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1,\ldots,q-1\}^n$ symbol by symbol via a channel limited to at most $pn$ errors and/or $p^{*} n$ erasures. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the …
We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Re…
This paper introduces a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes, this method requires only that the codes are highly symmetric. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge to a number between 0 and 1, and the permutation group of each code is doubly transitive. This also provides a rare example in information theory where symmetry alone…
This work proves new results on the ability of binary Reed-Muller codes to decode from random errors and erasures. We obtain these results by proving improved bounds on the weight distribution of Reed-Muller codes of high degrees. Specifically, given weight $β\in (0,1)$ we prove an upper bound on the number of codewords of relative weight at most $β$. We obtain new results in two different settings: for weights $β< 1/2$ and for weights that are close to $1/2$. Our new bounds on the weight distribution imply that RM codes with $m$ variables and degree $γm$, for some explicit constant $γ$, achie…
We consider a 1-to-$K$ communication scenario, where a source transmits private messages to $K$ receivers through a broadcast erasure channel, and the receivers feed back strictly causally and publicly their channel states after each transmission. We explore the achievable rate region when we require that the message to each receiver remains secret - in the information theoretical sense - from all the other receivers. We characterize the capacity of secure communication in all the cases where the capacity of the 1-to-$K$ communication scenario without the requirement of security is known. As a…
This paper investigates fundamental properties of nonlinear binary codes by looking at the codebook matrix not row-wise (codewords), but column-wise. The family of weak flip codes is presented and shown to contain many beautiful properties. In particular the subfamily fair weak flip codes, which goes back to Berlekamp, Gallager, and Shannon and which was shown to achieve the error exponent with a fixed number of codewords $M$, can be seen as a generalization of linear codes to an arbitrary number of codewords. Based on the column-wise approach, the $r$-wise Hamming distance is introduced as a …
For any prime power $q$, Mori and Tanaka introduced a family of $q$-ary polar codes based on $q$~by~$q$ Reed-Solomon polarization kernels. For transmission over a $q$-ary erasure channel, they also derived a closed-form recursion for the erasure probability of each effective channel. In this paper, we use that expression to analyze the finite-length scaling of these codes on the $q$-ary erasure channel with erasure probability $ε\in(0,1)$. Our primary result is that, for any $γ>0$ and $δ>0$, there is a $q_{0}$ such that, for all $q\geq q_{0}$, the fraction of effective channels with erasure ra…
The two-receiver broadcast packet erasure channel with feedback and memory is studied. Memory is modeled using a finite-state Markov chain representing a channel state. Two scenarios are considered: (i) when the transmitter has causal knowledge of the channel state (i.e., the state is visible), and (ii) when the channel state is unknown at the transmitter, but observations of it are available at the transmitter through feedback (i.e., the state is hidden). In both scenarios, matching outer and inner bounds on the rates of communication are derived and the capacity region is determined. It is s…
In this paper, we show that the Hadamard matrix acts as an extractor over the reals of the Renyi information dimension (RID), in an analogous way to how it acts as an extractor of the discrete entropy over finite fields. More precisely, we prove that the RID of an i.i.d. sequence of mixture random variables polarizes to the extremal values of 0 and 1 (corresponding to discrete and continuous distributions) when transformed by a Hadamard matrix. Further, we prove that the polarization pattern of the RID admits a closed form expression and follows exactly the Binary Erasure Channel (BEC) polariz…
We consider the transmission of a Gaussian vector source over a multi-dimensional Gaussian channel where a random or a fixed subset of the channel outputs are erased. Within the setup where the only encoding operation allowed is a linear unitary transformation on the source, we investigate the MMSE performance, both in average, and also in terms of guarantees that hold with high probability as a function of the system parameters. Under the performance criterion of average MMSE, necessary conditions that should be satisfied by the optimal unitary encoders are established and explicit solutions …
A polar coding scheme for fading channels is proposed in this paper. More specifically, the focus is Gaussian fading channel with a BPSK modulation technique, where the equivalent channel could be modeled as a binary symmetric channel with varying cross-over probabilities. To deal with variable channel states, a coding scheme of hierarchically utilizing polar codes is proposed. In particular, by observing the polarization of different binary symmetric channels over different fading blocks, each channel use corresponding to a different polarization is modeled as a binary erasure channel such th…
Polar codes are the latest breakthrough in coding theory, as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of discrete memoryless channels. Arıkan's polar encoder and successive cancellation decoder have complexities of $N \log N$, for code length $N$. Although, the complexity bound of $N \log N$ is asymptotically favorable, we report in this work methods to further reduce the encoding and decoding complexities of polar coding. The crux is to relax the polarization of certain bit-channels without performance degradation. We consider …
We prove that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within $ε> 0$ of the Shannon capacity with a block length, construction complexity, and decoding complexity all bounded by a {\em polynomial} in $1/ε$. Polar coding gives the {\em first known explicit construction} with rigorous proofs of all these properties; previous constructions were not known to achieve capacity with less than $\exp(1/ε)$ decoding complexity except for erasure channels. We establish the capacity-achieving property of polar codes via a direct analysis of th…
Luby Transform (LT) codes are a class of fountain codes that have proved to perform very efficiently over the erasure channel. These codes are rateless in the sense that an infinite stream of encoded symbols can be generated on the fly. Furthermore, every encoded symbol is information additive and can contribute in the decoding process. An important application of fountain codes which is being considered is the delivery of content over mobile wireless channels. Fountain codes have low computational complexity and fast encoding and decoding algorithms which makes them attractive for real time a…
In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1\}^n$ bit by bit via a channel limited to at most $pn$ corruptions. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th bit or not based on its view so far, i.e., its decision depends only on the transmitted bits $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has a full …
Herein, we focus on explicit constructions of $\ell\times\ell$ binary kernels with small scaling exponent for $\ell \le 64$. In particular, we exhibit a sequence of binary linear codes that approaches capacity on the BEC with quasi-linear complexity and scaling exponent $μ< 3$. To the best of our knowledge, such a sequence of codes was not previously known to exist. The principal challenges in establishing our results are twofold: how to construct such kernels and how to evaluate their scaling exponent. In a single polarization step, an $\ell\times\ell$ kernel $K_\ell$ transforms an underlying…
Constructing efficient low-rate error-correcting codes with low-complexity encoding and decoding have become increasingly important for applications involving ultra-low-power devices such as Internet-of-Things (IoT) networks. To this end, schemes based on concatenating the state-of-the-art codes at moderate rates with repetition codes have emerged as practical solutions deployed in various standards. In this paper, we propose a novel mechanism for concatenating outer polar codes with inner repetition codes which we refer to as polar coded repetition. More specifically, we propose to transmit a…
For the upcoming 6G wireless networks, a new wave of applications and services will demand ultra-high data rates and reliability. To this end, future wireless systems are expected to pave the way for entirely new fundamental air interface technologies to attain a breakthrough in spectrum efficiency (SE). This article discusses a new paradigm, named generalized polarization transform (GPT), to achieve an integrated design of coding, modulation, multi-antenna, multiple access, etc., in a real sense. The GPT enabled air interface develops far-reaching insights that the joint optimization of criti…
An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arıkan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (second) part provides a foundation of polarization theory based on the ergodic theory of binary operations which we developed in the first part. We show that a binary operation is polarizing if and only if it is uniformity preserving and its ri…
This paper presents a polar coding scheme to achieve secrecy in block fading binary symmetric wiretap channels without the knowledge of instantaneous channel state information (CSI) at the transmitter. For this model, a coding scheme that hierarchically utilizes polar codes is presented. In particular, on polarization of different binary symmetric channels over different fading blocks, each channel use is modeled as an appropriate binary erasure channel over fading blocks. Polar codes are constructed for both coding over channel uses for each fading block and coding over fading blocks for cert…
An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arıkan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (first) part of the paper introduces the mathematical framework that we will use in the second part to characterize the polarizing operations. We define uniformity preserving, irreducible, ergodic and strongly ergodic operations and we study the…
A simple model system is introduced for demonstrating how a single photon source might be used to transduce classical analog information. The theoretical scheme results in measurements of analog source samples that are (i) quantized in the sense of analog-to-digital conversion and (ii) corrupted by random noise that is solely due to the quantum uncertainty in detecting the polarization state of each photon. This noise is unavoidable if more than one bit per sample is to be transmitted, and we show how it may be exploited in a manner inspired by suprathreshold stochastic resonance. The system i…
