Binary-input Discrete Memoryless Channels Research Articles

Covert Communication With Mismatched Decoders

This paper considers the problem of covert communication over binary-input discrete memoryless channels with mismatched decoding, in which a sender wishes to reliably communicate with a receiver whose decoder is fixed and possibly sub-optimal, and simultaneously to ensure that the communication is covert with respect to a warden. We present a single-letter lower bound and two single-letter upper bounds on the information-theoretically optimal throughput as a function of the given decoding metric, channel laws, and the desired level of covertness. These bounds match for a variety of scenarios of interest, such as (i) when the channel between the sender and receiver is a binary-input binary-output channel, and (ii) when the decoding metric is particularized to the so-called erasures-only metric. The lower bound is obtained based on a modified random coding union bound with pulse position modulation (PPM) codebooks, coupled with a non-standard expurgation argument. The upper bounds are based on the idea of translating the mismatched-decoding error of the original channel to the decoding error of an auxiliary channel.

Optimal Rates of Teaching and Learning Under Uncertainty

In this paper, we consider a recently-proposed model of teaching and learning under uncertainty, in which a teacher receives independent observations of a single bit corrupted by binary symmetric noise, and sequentially transmits to a student through another binary symmetric channel based on the bits observed so far. After a given number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> of transmissions, the student outputs an estimate of the unknown bit, and we are interested in the exponential decay rate of the error probability as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> increases. We propose a novel block-structured teaching strategy in which the teacher encodes the number of 1s received in each block, and show that the resulting error exponent is the binary relative entropy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D\left({\frac {1}{2}\|\max (p,q)}\right)$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> are the noise parameters. This matches a trivial converse result based on the data processing inequality, and settles two conjectures of [Jog and Loh, 2021] and [Huleihel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2019]. In addition, we show that the computation time required by the teacher and student is linear in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> . We also study a more general setting in which the binary symmetric channels are replaced by general binary-input discrete memoryless channels. We provide an achievability bound and a converse bound, and show that the two coincide in certain cases, including (i) when the two channels are identical, and (ii) when the student-teacher channel is a binary symmetric channel. More generally, we give sufficient conditions under which our learning rate is the best possible for block-structured protocols.

Optimized Polar Codes as Forward Error Correction Coding for Digital Video Broadcasting Systems

Polar codes are featured by their low encoding/decoding complexity for symmetric binary input-discrete memoryless channels. Recently, flexible generic Successive Cancellation List (SCL) decoders for polar codes were proposed to provide different throughput, latency, and decoding performances. In this paper, we propose to use polar codes with flexible fast-adaptive SCL decoders in Digital Video Broadcasting (DVB) systems to meet the growing demand for more bitrates. In addition, they can provide more interactive services with less latency and more throughput. First, we start with the construction of polar codes and propose a new mathematical relation to get the optimized design point for the polar code. We prove that our optimized design point is too close to the one that achieves minimum Bit Error Rate (BER). Then, we compare the performance of polar and Low-Density Parity Check (LDPC) codes in terms of BER, encoder/decoder latencies, and throughput. The results show that both channel coding techniques have comparable BER. However, polar codes are superior to LDPC in terms of decoding latency, and system throughput. Finally, we present the possible performance enhancement of DVB systems in terms of decoding latency and complexity when using optimized polar codes as a Forward Error Correction (FEC) technique instead of Bose Chaudhuri Hocquenghem (BCH) and LDPC codes that are currently adopted in DVB standards.

Covert Identification Over Binary-Input Discrete Memoryless Channels

This paper considers the covert identification problem in which a sender aims to reliably convey an identification (ID) message to a set of receivers via a binary-input discrete memoryless channel (BDMC), and simultaneously to guarantee that the communication is covert with respect to a warden who monitors the communication via another independent BDMC. We prove a square-root law for the covert identification problem. This states that an ID message of size exp(exp(Θ(√{ n})) can be transmitted over n channel uses. We then characterize the exact pre-constant in the Θ(·) notation. This constant is referred to as the covert identification capacity. We show that it equals the recently developed covert capacity in the standard covert communication problem, and somewhat surprisingly, the covert identification capacity can be achieved without any shared key between the sender and receivers. The achievability proof relies on a random coding argument with pulse-position modulation (PPM), coupled with a second stage which performs code refinements. The converse proof relies on an expurgation argument as well as results for channel resolvability with stringent input constraints.

Error correction in data storage systems using polar codes

This paper investigates error correction in two-dimensional (2-D) intersymbol interference (ISI) channels using polar codes. 2-D channels offer high data storage capacity compared to traditional one-dimensional (1-D) channels but suffer from greater ISI effects. Error correction codes can be used to mitigate the effects of ISI in 2-D data storage systems and improve their performance and reliability. Polar codes achieve the symmetric capacity for the class of binary-input discrete memoryless channels (B-DMCs). A post-processing technique is proposed to improve the performance of polar codes in the 2-D ISI channel. The simplified successive cancellation (SSC) polar decoder is useful for data storage systems owing to its good waterfall region performance, low error floors and good throughput rates and is therefore adopted for the study. Simulation results indicate that the proposed technique has an error correction performance comparable to some existing decoders and has better performance than other decoding techniques.

A Recursive Quantizer Design Algorithm for Binary-Input Discrete Memoryless Channels

The optimal quantization of output binary-input discrete memoryless channels is considered, whereby the optimal quantizer preserves at least a constant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -fraction of the original mutual information, with the smallest output cardinality. Two recursive methods with top-down and bottom-up approaches are developed; these methods lead to a new necessary condition for the recursive quantizer design. An efficient algorithm with linear complexity, based on dynamic programming and the new necessary optimality condition, is proposed.

Capacity-Approaching Polar Codes With Long Codewords and Successive Cancellation Decoding Based on Improved Gaussian Approximation

This paper focuses on an improved Gaussian approximation (GA) based construction of polar codes with successive cancellation (SC) decoding over an additive white Gaussian noise (AWGN) channel. Arıkan proved that polar codes with low-complexity SC decoding can approach the channel capacity of an arbitrary symmetric binary-input discrete memoryless channel, provided that the code length is chosen large enough. Nevertheless, how to construct such codes over an AWGN channel with low computational effort has been an open problem. Compared to density evolution, the GA is known as a low complexity yet powerful technique that traces the evolution of the mean log likelihood ratio (LLR) value by iterating a nonlinear function. Therefore, its high-precision numerical evaluation is critical as the code length increases. In this work, by analyzing the asymptotic behavior of this nonlinear function, we propose an improved GA approach that makes an accurate trace of mean LLR evolution feasible. With this improved GA, through numerical analysis and simulations with code lengths up to N = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">18</sup> , we explicitly demonstrate that various code-rate polar codes with long codeword and capacity approaching behavior can be easily designed.

Multilevel-Coded Pulse-Position Modulation for Covert Communications Over Binary-Input Discrete Memoryless Channels

We consider the problem of coding to ensure covert communication, which involves ensuring reliable communication between two legitimate parties while simultaneously guaranteeing a low probability of detection by an eavesdropper. Specifically, we develop an optimal low-complexity coding scheme that achieves the information-theoretic limits of covert communications over binary-input discrete memoryless channels (BI-DMCs). To justify our design, we first consider a regime in which information theory proves the possibility of covert communication without shared secret key and show the impossibility of achieving information-theoretic limits using linear codes without secret key. We then circumvent this impossibility by introducing non-linearity into the coding scheme through the use of pulse position modulation (PPM) and multilevel coding (MLC). This MLC-PPM scheme exhibits several appealing properties; in particular, for an appropriate decoder, the channel at a given level is independent of the total number of levels and the codeword length. We exploit these properties to show how one can use families of channel capacity- and channel resolvability-achieving codes to concretely instantiate a covert communication scheme.

Performance Analysis of CRC-Polar Concatenated Codes

Polar code has been proven to obtain Shannon capacity for Binary Input Discrete Memoryless Channel (BIDMC) and its use has been proposed as the channel coding in 5G technology. However, its performance is limited in finite block length, compared to Turbo or LDPC codes. This research proposes the use of various CRC codes to complement Polar codes with finite block length and analyses the performance based on Block Error Rate (BLER) to Es/N0 (dB). The CRC codes used are of degrees 11 and 24, with 3 different polynomial generators for each degree. The number of bits in the information sequence is 32. The list sizes used are 1, 2, 4, and 8. Simulation results show that the concatenation of CRC and Polar codes will yield good BLER vs Es/N0 performance for short blocks of codeword, with rates 32/864 and 54/864. Concatenating CRC codes with Polar codes will yield a BLER performance of 10-2 with Es/N0 values of -9.1 to -7.5 dB when CRC codes of degree 11 is used, depending on the SC list used. The use of CRC codes of degree 24 enables a BLER performance of 10-2 with Es/N0 values of -7 to -6 dB when the SC list used is 1 or 2. The use of CRC codes of degree 24 combined with SC list with sizes 4 or 8 will improve the BLER performance to 10-2 with Es/N0 values of -8 to -7.5 dB

Polar Codes for Fast Fading Channel: Design Based on Polar Spectrum

As the first class of constructive channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels, polar codes have been extended to the fading channel. But most constructions of polar codes in fading channel are only suitable for the Rayleigh fading and involve complex iterative-calculation. In this paper, we establish a systematic framework in term of the polar spectrum to analyze and construct polar codes in various fast fading channels, such as Rician, Rayleigh and Nakagami channels. In these three fading channels, by using the polar spectrum, we derive the upper bound of the error probability of the polarized channel and that of block error rate of polar codes. The analysis based on the polar spectrum explicitly reveals the relationship between the diversity order and the codeword weight. Furthermore, we propose two construction metrics, named logarithmic upper-bound weight (LUW) and minimum-weight LUW (MLUW) respectively, to design the polar codes in fast fading channels. These two constructions are simple with a linear complexity and explicit for the practical implementation. In addition, for various fading channels, they can construct polar codes with similar or better performance over those based on traditional methods.

Stochastic Belief Propagation Polar Decoding With Efficient Re-Randomization

Polar codes, the first class of codes achieving the capacity of symmetric binary-input discrete memoryless channels (B-DMCs), have drawn intensive attentions. For speed-insensitive applications, such as implantable devices, there is a good need on complexity and robustness. In this article, stochastic computing is introduced to allieviate the contradiction between complexity and performance for polar decoders. We proposed a stochastic belief propagation (BP) polar decoder and corresponding novel approaches to improve its decoding performance. Based on the understanding of stochastic computing correlation (SCC) in polar coding, the efficient stage-wise re-randomization scheme and its implementation are proposed and carefully analyzed. Hardware architectural optimizations and corresponding ASIC implementation are also given and demonstrate advantages over the state-of-the-art.

An Efficient Software Stack Sphere Decoder for Polar Codes

Proved to achieve the symmetric capacity of the binary-input discrete memoryless channels, polar codes have been chosen for the eMBB control channels in the 5th generation mobile communication systems. Besides the main decoding algorithms like successive cancellation (SC) decoding and CRC-aid SC list (CA-SCL) decoding, sphere decoder (SD) and list SD (LSD) are the alternatives for short codes with less required memory bits. Existing SD and LSD attain high calculation complexity, for SD requires a back-tracking process and LSD needs a large list size $L$ to achieve satisfying performance. To reduce complexity, an efficient software stack sphere decoder (ESSD) based on the synchronous determination is firstly proposed in this article. With the dynamic set-by-set decoding in the stack structure, it achieves the lowest complexity in SD-based decoders (SD/LSD/ESSD) while sharing the same performance on low-rate codes and high-rate codes. Compared with the CA-SCL decoder, the complexity and latency of the proposed ESSD are also competitive at high signal-to-noise-ratio on the displayed codes. Implemented on C++ , the proposed ESSD reduces $\text{44.77}\%$ latency compared with CA-SCL-32 for $\mathcal {P}(128,120)$ at the BER of $10^{-5}$ with $E_b/N_0 = \text{7}$ dB.

A simplified decoding algorithm for multi-CRC polar codes

Polar codes represent one of the major breakthroughs in 5G standard, and have been proven to be able to achieve the symmetric capacity of binary-input discrete memoryless channels using the successive cancellation list (SCL) decoding algorithm. However, the SCL algorithm suffers from a large amount of memory overhead. This paper proposes an adaptive simplified decoding algorithm for multiple cyclic redundancy check (CRC) polar codes. Simulation results show that the proposed method can reduce the decoding complexity and memory space. It can also acquire the performance gain in the low signal to noise ratio region.

An Efficient Software List Sphere Decoder for Polar Codes

Polar codes, first achieving the capacity of symmetric binary-input discrete memoryless channels (B-DMCs), have been standardized for eMBB control channels. Since 5G cellular requires flexible architecture which is realized by the software defined networking paradigm, efficient polar decoder is anticipated. Though successive cancellation list (SCL) decoder achieves satisfactory performance, it requires a large amount of memory. For short control channel codes, sphere decoder (SD) is an alternative, but costs unbearable time complexity at low signal-to-noise ratio. List sphere decoder (LSD) abandons the radius and keeps a list of best paths to gain a fixed complexity. However, LSD needs a large list size L for satisfactory performance. In this paper, an efficient software LSD with path pruning and efficient sorting is proposed. We recall the radius as the bound to delete the paths out of the sphere at very early levels. Since L is dynamic, efficient sorting is proposed to reduce the copy operations. Implemented with C++, the proposed decoder can reduce up to 65.3% latency compared with the original LSD, with the same performance and lower complexity.

Covert Communication With Polynomial Computational Complexity

This paper develops a concatenated coding scheme with polynomial computational complexity for covert communication over Binary Symmetric Channels (BSCs) and binary-input Discrete Memoryless Channels (DMCs). Our setting is as follows — a transmitter Alice wishes to potentially reliably transmit a message to a receiver Bob, while ensuring that the transmission taking place is covert with respect to a warden Willie (who hears Alice’s transmission over another independent channel). Prior works showed that Alice can reliably and covertly transmit $\mathcal {O}(\sqrt {{n}})$ message bits over n channel uses, but one drawback is that the computational complexity of the codes designed scales as $2^{\Theta (\sqrt {{n}})}$ . In this work we provide a capacity-achiveing coding scheme with provable guarantees on both reliability and covertness, and its computational complexity grows polynomially in the blocklength n .

Embedding Covert Information in Broadcast Communications

We analyze a two-receiver binary-input discrete memoryless broadcast channel, in which the transmitter communicates a common message simultaneously to both receivers and a covert message to only one of them. The unintended recipient of the covert message is treated as an adversary who attempts to detect the covert transmission. This model captures the problem of embedding covert messages in an innocent codebook and generalizes previous covert communication models in which innocent behavior corresponds to the absence of communication between legitimate users. We identify the exact asymptotic behavior of the number of covert bits that can be transmitted when the rate of the innocent codebook is close to the capacity of the channel to the adversary. Our results also identify the dependence of the number of covert bits on the channel parameters and the characteristics of the innocent codebook.

An Improved Software List Sphere Polar Decoder With Synchronous Determination

Being the first codes achieving the capacity of symmetric binary-input discrete memoryless channels, polar codes have been favorable and standardized for enhanced mobile broadband (eMBB) control channels. In future mobile communications, the entire system is required to be flexible and adaptive to varying scenarios. Therefore, efficient software implementation of polar decoder is expected. Existing software successive cancellation list polar decoder can achieve good performance but with a good memory consumption. For short control channel codes, a more memory-efficient alternative is sphere decoder, but its latency gives rise to another issue. Equipped with a breadth-first search to keep a fixed complexity, list sphere decoder (LSD) is also considered. To improve the performance of LSD, a software LSD with synchronous determination is proposed in this paper. We precalculate the synchro sets to improve the decoding, which leads to a performance gain. Meanwhile, the decoding process is simplified due to the precalculation, which attains even lower complexity. Implemented with C++ , the proposed decoder can achieve up to 2 dB performance gain compared with the existing LSD.

Segmented Successive Cancellation List Polar Decoding with Tailored CRC

As the first error correction codes provably achieving the symmetric capacity of binary-input discrete memory-less channels (B-DMCs), polar codes have been recently chosen by 3GPP for eMBB control channel. Among existing algorithms, CRC-aided successive cancellation list (CA-SCL) decoding is favorable due to its good performance, where CRC is placed at the end of the decoding and helps to eliminate the invalid candidates before final selection. However, the good performance is obtained with a complexity increase that is linear in list size $L$. In this paper, the tailored CRC-aided SCL (TCA-SCL) decoding is proposed to balance performance and complexity. Analysis on how to choose the proper CRC for a given segment is proposed with the help of \emph{virtual transform} and \emph{virtual length}. For further performance improvement, hybrid automatic repeat request (HARQ) scheme is incorporated. Numerical results have shown that, with the similar complexity as the state-of-the-art, the proposed TCA-SCL and HARQ-TCA-SCL schemes achieve $0.1$ dB and $0.25$ dB performance gain at frame error rate $\textrm{FER}=10^{-2}$, respectively. Finally, an efficient TCA-SCL decoder is implemented with FPGA demonstrating its advantages over CA-SCL decoder.

Efficient Implementation of Density Evolution for Punctured Polar Codes

Polar codes asymptotically achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under low-complexity sequential decoding algorithms such as successive cancellation decoding. However, in their original formulation, the block length of polar codes is limited to integer powers of the dimension of the underlying polarization kernel used, thus imposing strict constraints on possible application scenarios. While leeway in the choice of kernel or concatenation with other codes mitigates this drawback to a certain extent, puncturing presents a promising approach to specify the target length of a polar code with much greater flexibility. In this paper, we present an efficient implementation of the construction of punctured polar codes based on density evolution, a crucial tool in the construction of both regular, i.e., unpunctured, as well as punctured polar codes. Our implementation of density evolution covers the construction of both regular and punctured polar codes and allows for treating the construction of both code classes in a unified framework. Using our implementation, we achieve substantial reductions in the number of density convolutions necessary for the construction of punctured polar codes and obtain tight upper bounds on the block error rates.

Polar Codes: Bounds on Bhattacharyya Parameters and Their Applications

This paper proposes a novel lower bound on the Bhattacharyya parameter of synthesized channels in polar codes. The proposed bound is a result of a newly defined merge operation that reduces the cardinality of output alphabets of a symmetric binary-input discrete memoryless channel. By repeated applications of the proposed merge operation, we establish a universal bound on the Bhattacharyya parameters of synthesized channels that holds for any symmetric binary-input discrete memoryless transmission channel. We then discuss some applications of the new bound. For example, using the derived universal lower bound along with the existing upper bound, we find a partial order among the Bhattacharyya parameters of those synthesized channels that cannot be ordered through existing results. This ordering can help efficient design of polar codes. In addition, some extremal behaviors of the binary-symmetric channel are also proved.