Quantum reading provides a general framework where to formulate the statistical discrimination of quantum channels. Several paths have been taken for such a problem. However, there is much to be done in the avenue of optimizing channel discrimination using classical codes. At least two open questions can be pointed to: how to construct low complexity encoding schemes that are interesting for channel discrimination and, more importantly, how to develop capacity-achieving protocols. The aim of this paper is to present a solution to these questions using polar codes. Firstly, we characterize the rate and reliability of the channels under polar encoding. We also show that the error probability of the scheme proposed decays exponentially with respect to the code length. Lastly, an analysis of the optimal quantum states to be used as probes is given.
A general framework describing the statistical discrimination of an ensemble of quantum channels is given by the name of quantum reading. Several tools can be applied in quantum reading to reduce the error probability in distinguishing the ensemble of channels. Classical and quantum codes can be envisioned for this goal. The aim of this paper is to present a simple but fruitful protocol for this task using classical error-correcting codes. Three families of codes are considered: Reed-Solomon codes, BCH codes, and Reed-Muller codes. In conjunction to the use of codes, we also analyze the role o…
This paper presents a practical writing/reading scheme in nonvolatile memories, called balanced modulation, for minimizing the asymmetric component of errors. The main idea is to encode data using a balanced error-correcting code. When reading information from a block, it adjusts the reading threshold such that the resulting word is also balanced or approximately balanced. Balanced modulation has suboptimal performance for any cell-level distribution and it can be easily implemented in the current systems of nonvolatile memories. Furthermore, we studied the construction of balanced error-corre…
The adversarial wiretap channel (AWTC) model is a secure communication model in which adversary can directly read and write the transmitted bits in legitimate communication with fixed fractions. In this paper we propose a secure polar coding scheme to provide secure and reliable communication over the AWTC model. For the adversarial reading and writing action, we present a $ρ$ equivalent channel block and study its transformation under the channel polarization operation. We find that the generated channels are polarized in the sense that part of them are full-noise channels with probability al…
An important practical open question has been to design explicit, structured optical receivers that achieve the Holevo limit in the contexts of optical communication and "quantum reading." The Holevo limit is an achievable rate that is higher than the Shannon limit of any known optical receiver. We demonstrate how a sequential decoding approach can achieve the Holevo limit for both of these settings. A crucial part of our scheme for both settings is a non-destructive "vacuum-or-not" measurement that projects an n-symbol modulated codeword onto the n-fold vacuum state or its orthogonal compleme…
Flash memories intended for SSD and mobile applications need to provide high random I/O performance. This requires using efficient schemes for reading small chunks of data (e.g. 0.5KB - 4KB) from random addresses. Furthermore, in order to be cost efficient, it is desirable to use high density Multi-Level Cell (MLC) memories, such as the ones based on 3 or 4 bit per cell technologies. Unfortunately, these two requirements are contradicting, as reading an MLC memory, whose data is coded conventionally, requires multiple sensing operations, resulting in slow reading and degraded random I/O perfor…
Arikan's polar codes are capable of achieving the Shannon's capacity at a low encoding and decoding complexity, while inherently supporting rate adaptation. By virtue of these attractive features, polar codes have provided fierce competition to both the turbo as well as the Low Density Parity Check (LDPC) codes, making its way into the 5G New Radio (NR). Realizing the significance of polar codes, in this paper we provide a comprehensive survey of polar codes, highlighting the major milestones achieved in the last decade. Furthermore, we also provide tutorial insights into the polar encoder, de…
Polar codes, introduced by Arikan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes u…
In this letter we propose a new hybrid code called "RM-Polar" codes. This new codes are constructed by combining the construction of Reed-Muller (RM) code and Polar code. It has much larger minimum Hamming distance than Polar codes, therefore it has much better error performance than Polar codes.
In this paper, we study polar codes from a practical point of view. In particular, we study concatenated polar codes and rate-compatible polar codes. First, we propose a concatenation scheme including polar codes and Low-Density Parity-Check (LDPC) codes. We will show that our proposed scheme outperforms conventional concatenation schemes formed by LDPC and Reed-Solomon (RS) codes. We then study two rate-compatible coding schemes using polar codes. We will see that polar codes can be designed as universally capacity achieving rate-compatible codes over a set of physically degraded channels. We…
In this paper, we study the connection between polar codes and product codes. Our analysis shows that the product of two polar codes is again a polar code, and we provide guidelines to compute its frozen set on the basis of the frozen sets of the component polar codes. Moreover, we show how polar codes can be described as irregular product codes. We propose a two-step decoder for long polar codes taking advantage of this dual nature to heavily reduce decoding latency. Finally, we show that the proposed decoding technique outperforms both standard polar codes and state-of-the-art codes for opti…
In this paper, we present an iterative construction of a polar code and develop properties of the dual of a polar code. Based on this approach, belief propagation of a polar code can be presented in the context of low-density parity check codes.
We consider the problem of supporting H-ARQ with polar codes. For supporting H-ARQ, we propose to create redundancy versions based on different, but equivalent, subsets of a polar code. The equivalent subsets are created from an initial subset of a polar code using an inherent symmetry in polar code construction. A greedy construction is used to create the initial subset of a polar code. We demonstrate performance of proposed constructions via simulations for binary input AWGN channel. We demonstrate that a (4096, 1024) polar code can be divided into two disjoint (2048, 1024) subset polar code…
We survey the known list decoding algorithms for polar codes and compare their complexity. Index terms: Polar codes; Reed-Muller codes; successive cancellation decoding.
Polar codes under successive cancellation decoding proposed by Arıkan provably achieve the symmetric capacity of any given binary-input discrete memoryless channel. The successive cancellation list decoder for polar codes was described by Tal and Vardy as a generalization of the successive cancellation decoder of Arıkan. The performance of the successive cancellation list decoder is encouraging in practice. In this paper, we formalize the successive cancellation list decoder in our notation and prove that polar codes under successive cancellation list decoding achieve the symmetric capacity of…
By inserting interleavers between intermediate stages of the polar encoder, a new class of polar codes, termed interleaved polar (i-polar) codes, is proposed. By the uniform interleaver assumption, we derive the weight enumerating function (WEF) and input-output weight enumerating function (IOWEF) averaged over the ensemble of i-polar codes. The average WEF can be used to calculate the upper bound on the average block error rate (BLER) of a code selected at random from the ensemble of i-polar codes. Also, we propose a concatenated coding scheme that employs P high rate codes as the outer code …
Polar codes are considered the latest major breakthrough in coding theory. Polar codes were introduced by Arıkan in 2008. In this letter, we show that the binary polar codes are the same as the optimized codes for bitwise multistage decoding (OCBM), which have been discovered before by Stolte in 2002. The equivalence between the techniques used for the constructions and decodings of both codes is established.
In this paper, we propose a new class of nonbinary polar codes with two-stage polarization for ultra-reliable and low-latency communications (URLLC), where the outer (symbol-level) polarization is achieved by using a 2X2 q-ary matrix [1 0; β1] as the kernel and the inner (bit-level) polarization is achieved by a binary polarization matrix for each input symbol. With the two-stage polarization, bit-level code construction is introduced, resulting in partially-frozen symbols, where the frozen bits in these symbols can be used as active-check bits to facilitate the decoder. The encoder/decoder of…
Polar codes are the first proven capacity-achieving codes. Recently, they are adopted as the channel coding scheme for 5G due to their superior performance.A polar code for encoding length-K information bits in length-N codeword could be specified by the polar code construction method. Most construction methods define a polar code related to channel parameter set, e.g. designed signal-to-noise ratio. Polarization weight (PW) is a channel-independent approximation method, which estimates the subchannel reliability as a function of its index. In this paper, we generalize the PW method by includi…
Polar codes are the first class of constructive channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels. But the analysis and construction of polar codes involve the complex iterative-calculation. In this paper, by revisiting the error event of the polarized channel, a new concept, named polar spectrum, is introduced from the weight distribution of polar codes. Thus we establish a systematic framework in term of the polar spectrum to analyze and construct polar codes. By using polar spectrum, we derive the union bound and the union-Bhattacharyya (UB) bou…
Code decompositions (a.k.a code nestings) are used to design good binary polar code kernels. The proposed kernels are in general non-linear and show a better rate of polarization under successive cancelation decoding, than the ones suggested by Korada et al., for the same kernel dimensions. In particular, kernels of sizes 14, 15 and 16 are constructed and shown to provide polarization rates better than any binary kernel of such sizes.
This paper presents a polarization-driven (PD) shortening technique for the design of rate-compatible polar codes. The proposed shortening strategy consists of reducing the generator matrix by relating its row index with the channel polarization index. We assume that the shortened bits are known by both the encoder and the decoder and employ successive cancellation (SC) for decoding the shortened codes constructed by the proposed PD technique. A performance analysis is then carried out based on the Spectrum Distance (SD). Simulations show that the proposed PD-based shortened polar codes outper…
We propose a novel scheme for rate-compatible arbitrary-length polar code construction for the additive white Gaussian noise (AWGN) channel. The proposed scheme is based on the concept of non-uniform channel polarization. The original polar codes can only be designed with code lengths that are powers of two. Puncturing, shortening and extension are three strategies to obtain arbitrary code lengths and code rates for polar codes. There are other ways to design codes with arbitrary length but which have encoding and decoding with higher complexity such as multi-kernel, concatenated codes and spe…
Symbol-pair codes, introduced by Cassuto and Blaum [1], have been raised for symbol-pair read channels. This new idea is motivated by the limitation of the reading process in high-density data storage technologies. Yaakobi et al. [8] introduced codes for $b$-symbol read channels, where the read operation is performed as a consecutive sequence of $b>2$ symbols. In this paper, we come up with a method to compute the $b$-symbol-pair distance of two $n$-tuples, where $n$ is a positive integer. Also, we deal with the $b$-symbol-pair distances of some kind of cyclic codes of length $p^e$ over $\math…
In this work we propose an encoding and decoding framework for staircase codes based on non-systematic polar codes as component codes. The staircase structure allows for efficient parallelized decoding, while the polar component codes allow to benefit from the flexible structure and efficient soft-decision decoding algorithms. To enhance the performance of the polar staircase codes, we concatenate the polar component codes with cyclic redundancy check (CRC) outer codes, and we add interleavers within the staircase structure that are specific to polar code properties. The CRCs also allow to sub…
We consider the problem of using polar codes with higher order modulation over AWGN channels. Unlike prior work, we focus on using modulation independent polar codes. That is, the polar codes are not re-designed based on the modulation used. Instead, we propose bit-permuted coded modulation (BPCM): a technique for using the multilevel coding (MLC) approach for an arbitrary polar code. The BPCM technique exploits a natural connection between MLC and polar codes. It involves applying bit permutations prior to mapping the polar code to a higher order modulation. The bit permutations are designed,…
A polar code extension method which supports incremental redundancy hybrid ARQ(IR-HARQ) is proposed in this paper. By copying information bits to proper positions of the extend part, the extended polar code can give additional protection for the bits weakly protected by the first transmission. A comparison between the proposed algorithm with directly generated polar code, LTE turbo code, and some other IR-HARQ supporting polar code is given. Simulation results show that the proposed algorithm has nearly the same performance as directly generated polar code, which is chosen as base line for com…
A reduced complexity sequential decoding algorithm for polar (sub)codes is described. The proposed approach relies on a decomposition of the polar (sub)code being decoded into a number of outer codes, and on-demand construction of codewords of these codes in the descending order of their probability. Construction of such codewords is implemented by fast decoding algorithms, which are available for many codes arising in the decomposition of polar codes. Further complexity reduction is achieved by taking hard decisions of the intermediate LLRs, and avoiding decoding of some outer codes. Data str…
Explicit coding schemes are proposed to achieve the rate-distortion function of the Heegard-Berger problem using polar codes. Specifically, a nested polar code construction is employed to achieve the rate-distortion function for the doubly-symmetric binary sources when the side information may be absent. The nested structure contains two optimal polar codes for lossy source coding and channel coding, respectively. Moreover, a similar nested polar lattice construction is employed when the source and the side information are jointly Gaussian. The proposed polar lattice is constructed by nesting …
Polar codes were introduced in 2009 by Arikan as the first efficient encoding and decoding scheme that is capacity achieving for symmetric binary-input memoryless channels. Recently, this code family was extended by replacing the block-structured polarization step of polar codes by a convolutional structure. This article presents a numerical exploration of this so-called convolutional polar codes family to find efficient generalizations of polar codes, both in terms of decoding speed and decoding error probability. The main conclusion drawn from our study is that increasing the convolution dep…
Product codes are widespread in optical communications, thanks to their high throughput and good error-correction performance. Systematic polar codes have been recently considered as component codes for product codes. In this paper, we present a novel construction for product polar codes based on non-systematic polar codes. We prove that the resulting product code is actually a polar code, having a frozen set that is dependent on the frozen sets of the component polar codes. We propose a low-complexity decoding algorithm exploiting the dual nature of the constructed code. Performance analysis …
We present a comparative study of the performance of various polar code constructions in an additive white Gaussian noise (AWGN) channel. A polar code construction is any algorithm that selects $K$ best among $N$ possible polar bit-channels at the design signal-to-noise-ratio (design-SNR) in terms of bit error rate (BER). Optimal polar code construction is hard and therefore many suboptimal polar code constructions have been proposed at different computational complexities. Polar codes are also non-universal meaning the code changes significantly with the design-SNR. However, it is not known w…
We consider two problems related to polar codes. First is the problem of polar codes construction and analysis of their performance without Monte-Carlo method. The formulas proposed are the same as those in [Mori-Tanaka], yet we believe that our approach is original and has clear advantages. The resulting computational procedure is presented in a fast algorithm form which can be easily implemented on a computer. Secondly, we present an original method of construction of concatenated codes based on polar codes. We give an algorithm for construction of such codes and present numerical experiment…
A heuristic construction of polar codes for successive cancellation list (SCL) decoding with a given list size is proposed to balance the trade-off between performance measured in frame error rate (FER) and decoding complexity. Furthermore, a construction based on dynamically frozen bits with constraints among the "low weight bits" (LWB) is presented. Simulation results show that the LWB-polar codes outperform the CRC-polar codes and the eBCH-polar codes under SCL decoding.
In this paper, we present the concept of non-uniform channel polarization and a scheme for rate-compatible polar code construction for any codeword length using additive white Gaussian noise (AWGN) channels and the successive cancellation (SC) decoder. A Non-Uniform Polarization technique based on the Gaussian Approximation (NUPGA) is developed and an efficient rate-compatible design technique is devised to choose the best channels for transmission by a process of re-polarization of the codeword with the desired length. Simulations illustrate the proposed NUPGA design against existing rate-com…
In this paper, a clustered wireless sensor network is considered that is modeled as a set of coupled Gaussian multiple-access channels. The objective of the network is not to reconstruct individual sensor readings at designated fusion centers but rather to reliably compute some functions thereof. Our particular attention is on real-valued functions that can be represented as a post-processed sum of pre-processed sensor readings. Such functions are called nomographic functions and their special structure permits the utilization of the interference property of the Gaussian multiple-access channe…
In this paper, we prove that any pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and convolution code (CC) matrix) does not reduce the code minimum distance and an properly designed pre-transformation can reduce the number of codewords with the minimum distance. This explains that the pre-transformed polar codes can perform better than the Polar/RM codes.
Systematic polar codes are shown to outperform non-systematic polar codes in terms of the bit-error-rate (BER) performance. However theoretically the mechanism behind the better performance of systematic polar codes is not yet clear. In this paper, we set the theoretical framework to analyze the performance of systematic polar codes. The exact evaluation of the BER of systematic polar codes conditioned on the BER of non-systematic polar codes involves in $2^{NR}$ terms where $N$ is the code block length and $R$ is the code rate, resulting in a prohibitive number of computations for large block…
The symbol-pair read channel was first proposed by Cassuto and Blaum. Later, Yaakobi et al. generalized it to the $b$-symbol read channel. It is motivated by the limitations of the reading process in high density data storage systems. One main task in $b$-symbol coding theory is to determine the $b$-symbol weight hierarchy of codes. In this paper, we study the $b$-symbol weight hierarchy of the Kasami codes, which are well known for their applications to construct sequences with optimal correlation magnitudes. The complete symbol-pair weight distribution of the Kasami codes is determined.
Recently, Arıkan introduced the method of channel polarization on which one can construct efficient capacity-achieving codes, called polar codes, for any binary discrete memoryless channel. In the thesis, we show that decoding algorithm of polar codes, called successive cancellation decoding, can be regarded as belief propagation decoding, which has been used for decoding of low-density parity-check codes, on a tree graph. On the basis of the observation, we show an efficient construction method of polar codes using density evolution, which has been used for evaluation of the error probability…
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 for this design. To select the rates of the Construction D component codes, rather than using the capacity as in past work, we use the explicit finite-length properties of the polar code. Under successive cancellation decoding, density evolution allows choosing code rates that satisfy the equal error probability rule. At an error-rate of $10^{-4}$, a dimension…
Polar codes are the first class of channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels with efficient encoding and decoding algorithms. But the weight spectrum of Polar codes is relatively poor compared to RM codes, which degrades their ML performance. Pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and polarization-adjusted convolutional (PAC) codes), improves weight spectrum while retaining polarization. In this paper, the weight spectrum of upper-triangular pre-transformed Polar Codes …
A scheme for concatenating the recently invented polar codes with non-binary MDS codes, as Reed-Solomon codes, is considered. By concatenating binary polar codes with interleaved Reed-Solomon codes, we prove that the proposed concatenation scheme captures the capacity-achieving property of polar codes, while having a significantly better error-decay rate. We show that for any $ε> 0$, and total frame length $N$, the parameters of the scheme can be set such that the frame error probability is less than $2^{-N^{1-ε}}$, while the scheme is still capacity achieving. This improves upon $2^{-N^{0.5-ε…
A commonly assumed drawback of multi-level coding, compared to a bit-interleaved coded modulation, is its high latency: Indeed, the levels must be decoded sequentially. In this paper, we consider polar codes to code each level. We show that the decoding time complexity of the multi-level scheme, using successive-cancellation list decoding for each polar code, is only 1.5 times the one of a single polar code, regardless of the signal-to-noise ratio and the number of levels.
A new family of codes based on polar codes, soft concatenation and list+CRC decoding is proposed. Numerical experiments show the performance competitive with industry standards and Tal, Vardy approach.
In this paper we deal with polar code automorphisms that are beneficial under low-latency automorphism ensemble (AE) decoding, and we propose polar code designs that have such automorphisms. Successive-cancellation (SC) decoding and thus SC-based AE decoding are invariant with respect to the only known polar code automorphisms, namely those of the lower-triangular affine (LTA) group. To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms. Building on this, we design specific polar codes that admit automo…
Polar codes are recursive general concatenated codes. This property motivates a recursive formalization of the known decoding algorithms: Successive Cancellation, Successive Cancellation with Lists and Belief Propagation. Using such description allows an easy development of these algorithms for arbitrary polarizing kernels. Hardware architectures for these decoding algorithms are also described in a recursive way, both for Arikan's standard polar codes and for arbitrary polarizing kernels.
Data availability (DA) attack is a well-known problem in certain blockchains where users accept an invalid block with unavailable portions. Previous works have used LDPC and 2-D Reed Solomon (2DRS) codes with Merkle trees to mitigate DA attacks. These codes perform well across various metrics such as DA detection probability and communication cost. However, these codes are difficult to apply to blockchains with large blocks due to large decoding complexity and coding fraud proof size (2D-RS codes), and intractable code guarantees for large code lengths (LDPC codes). In this paper, we focus on …
Polar codes have been shown to approach capacity of symmetric binary erasure channels and also have low encoding and decoding complexity. Wireless channels are bursty in nature and a method to apply polar codes over wireless channels is proposed. This paper investigates the performance of polar codes over typical urban fading wireless channels. The wireless channel is modeled as a degraded erasure channel by using an inner convolution code and interleaving. When the convolution code fails to correct any errors, the corresponding bits are discarded and treated as erasures. We present the perfor…
