This paper presents a novel methodology for designing the reliability sequence π for polarization-adjusted convolutional (PAC) and polar codes. Given a sequence π of length N, a family of nested information sets is constructed for N codes. The design objective is to minimize the average of the minimum signal-to-noise ratios (SNRs) required to achieve a target block error rate (BLER) across these nested codes. To reduce the search space, the N bit positions are partitioned into M components of length N/M, with bit positions within each component following a predefined reliability order. The design problem is then formulated as a Markov decision process (MDP) and solved using a deep Q-network (DQN) with an experience replay buffer. Simulation results under CRC-aided successive cancellation list (CA-SCL) decoding for nested polar codes show that the proposed sequence improves SNR performance over existing nested polar codes, including those used in 5G. Additionally, nested PAC codes under successive cancellation list (SCL) decoding with the proposed π outperform nested PAC and polar codes based on the known reliability sequences, under SCL or CA-SCL decoding. Moreover, individual PAC codes extracted from the nested family perform competitively with the best-known PAC/polar codes designed for specific rates. The proposed method is applicable to various precoded polar code constructions and decoding algorithms. The observed performance improvements, particularly in PAC codes, suggest that this approach is a promising candidate for future communication systems, including 6G.

This paper introduces a novel rate-profile design for Polarization-Adjusted Convolutional (PAC) codes by simplifying the construction of an information set (inf set) for a PAC code of length$N$into the design of inf sets for$M$component codes of length$\frac{N}{M}$. We define a Markov Decision Process (MDP) that formulates the task of identifying the inf set that minimizes the Block Error Rate (BLER). To achieve this, we employ a Deep QNetwork (DQN) with an experience replay buffer to determine the optimal inf set with the lowest BLER. Simulation results demonstrate that under Successive Cancellation List (SCL) decoding, our designed PAC codes improve BLER compared to state-of-the-art PAC codes. This method is not limited to the design of PAC code inf set; it is applicable to inf set design for the general family of precoded Polar codes and supports any decoder.

Turbo Autoencoder (TAE) is a deep-learning based channel code that demonstrates promising error correction performance. This paper studies the rate-matching problem for TAE and proposes a rate-matched TAE framework. The rate-matched TAE is a single auto-encoder model that can be used for multiple code rates. It matches the code rate of a mother$(3k, k)$TAE to a desired code of parameters ($n^{\ast}, k^{\ast}$) by using a combination of freezing message bits, repeating code symbols, and puncturing code symbols. We refer to the conventional TAE with message word length$k$and code length$3k$as mismatched TAE. The rate-matched TAE shares the same encoder and decoder structure with mismatched TAE but is trained to jointly optimize the performance across multiple rates. We study two important hyper-parameters for the rate-matched TAE: puncturing pattern and training signal-to-noise ratio (SNR) for constituent rates. Three puncturing patterns, namely, head, tail, and uniform puncturing are proposed and evaluated. Training SNRs are determined according to a heuristic method that uses test loss as a performance metric. Our simulation results show that the rate-matched TAE for$k= 100$and rates$r\in \{0.1, 0.2, \ldots, 0.9\}$significantly outperforms the mismatched TAE when$r\geq 0.4$.

This paper evaluates the performance of Turbo Autoencoder (TurboAE), an end-to-end jointly trained neural channel encoder and decoder, for different well-known inter-leavers over additive white Gaussian noise (AWGN) channel. TurboAE and Turbo codes share structural similarities. Therefore, in search of suitable interleavers for TurboAE, it is important to examine the interleavers designed for Turbo codes. In this paper, different interleavers are compared for TurboAE, and it is shown that using interleavers such as Quadratic Permutation Polynomial (QPP) and S-random interleavers provides considerable performance gain. The interleaver gain is achieved for both TurboAE made of convolutional neural network (CNN) layers and TurboAE made of recurrent neural networks (RNN) layers. Certain techniques are also proposed to enhance the performance of TurboAE RNN.

In this paper, we present list autoencoder (listAE) to mimic list decoding used in classical coding theory. With listAE, the decoder network outputs a list of decoded message word candidates. To train the listAE, a genie is assumed to be available at the output of the decoder. A specific loss function is proposed to optimize the performance of a genie-aided (GA) list decoding. The listAE is a general framework and can be used with any AE architecture. We propose a specific architecture, referred to as incremental-redundancy AE (IR-AE), which decodes the received word on a sequence of component codes with non-increasing rates. Then, the listAE is trained and evaluated with both IR-AE and Turbo-AE. Finally, we employ cyclic redundancy check (CRC) codes to replace the genie at the decoder output and obtain a CRC aided (CA) list decoder. Our simulation results show that the IR-AE under CA list decoding demonstrates meaningful coding gain over Turbo-AE and polar code at low block error rates range.

There have been significant research activities in recent years to automate the design of channel encoders and decoders via deep learning. Due the dimensionality challenge in channel coding, it is prohibitively complex to design and train relatively large neural channel codes via deep learning techniques. Consequently, most of the results in the literature are limited to relatively short codes having less than 100 information bits. In this paper, we construct ProductAEs, a computationally efficient family of deep-learning driven (encoder, decoder) pairs, that aim at enabling the training of relatively large channel codes (both encoders and decoders) with a manageable training complexity. We build upon the ideas from classical product codes, and propose constructing large neural codes using smaller code components. More specifically, instead of directly training the encoder and decoder for a large neural code of dimension k and blocklength n, we provide a framework that requires training neural encoders and decoders for the code parameters (n1,k1) and (n2,k2) such that n1n2= n and k1k2= k. Our training results show significant gains, over all ranges of signal-to-noise ratio (SNR), for a code of parameters (225,100) and a moderate-length code of parameters (441,196), over polar codes under successive cancellation (SC) decoder. Moreover, our results demonstrate meaningful gains over Turbo Autoencoder (TurboAE) and state-of-the-art classical codes. This is the first work to design product autoencoders and a pioneering work on training large channel codes.

This paper enhances the performance and training of the Turbo Autoencoder (TurboAE), an end-to-end jointly trained neural channel encoder and decoder. A novel interleaver for TurboAE with convolutional neural network (CNN) is proposed, which is shown to enhance the end-to-end performance of both TurboAE continuous and binary. Thanks to the proposed interleaver, TurboAE binary performs better than a benchmark Turbo code of length 100. In addition, a novel additional term to the loss function is proposed to improve the pairwise distance distribution of the codewords, which is shown to help the training convergence of TurboAE with CNN and recurrent neural network (RNN).

The performance of low-density parity-check (LDPC) codes at high signal-to-noise ratios (SNRs) is known to be limited by the presence of certain sub-graphs that exist in the Tanner graph representation of the code, for example trapping sets and absorbing sets. This paper derives a lower bound on the frame error rate (FER) of any LDPC code containing a given problematic sub-graph, assuming a particular message passing decoder and decoder quantization. A crucial aspect of the lower bound is that it is code-independent, in the sense that it can be derived based only on a problematic sub-graph and then applied to any code containing it. Due to the complexity of evaluating the exact bound, assumptions are proposed to approximate it, from which we can estimate decoder performance. Simulated results obtained for both the quantized sum-product algorithm (SPA) and the quantized min-sum algorithm (MSA) are shown to be consistent with the approximate bound and the corresponding performance estimates. Different classes of LDPC codes, including both structured and randomly constructed codes, are used to demonstrate the robustness of the approach.

For decoding low-density parity-check (LDPC) codes, the attenuated min-sum algorithm (AMSA) and the offset min-sum algorithm (OMSA) can outperform the conventional min-sum algorithm (MSA) at low signal-to-noise-ratios (SNRs), i.e., in the “waterfall region” of the bit error rate curve. This paper demonstrates that, for quantized decoders, MSA actually outperforms AMSA and OMSA in the “error floor” region, and that all three algorithms suffer from a relatively high error floor. This motivates the introduction of a modified MSA that is designed to outperform MSA, AMSA, and OMSA across all SNRs. The new algorithm is based on the assumption that trapping sets are the major cause of the error floor for quantized LDPC decoders. A performance estimation tool based on trapping sets is used to verify the effectiveness of the new algorithm and also to guide parameter selection. We also show that the implementation complexity of the new algorithm is only slightly higher than that of AMSA or OMSA. Finally, the simulated performance of the new algorithm, using several classes of LDPC codes (including spatially coupled LDPC codes), is shown to outperform MSA, AMSA, and OMSA across all SNRs.

It is well known that for decoding low-density parity-check (LDPC) codes, the attenuated min-sum algorithm (AMSA) and the offset min-sum algorithm (OMSA) can outperform the conventional min-sum algorithm (MSA) at low signal-to-noise-ratios (SNRs). In this paper, we demonstrate that, for quantized LDPC decoders, although the MSA achieves better high SNR performance than the AMSA and OMSA, each of the MSA, AMSA, and OMSA all suffer from a relatively high error floor. Therefore, we propose a novel modification of the MSA for decoding quantized LDPC codes with the aim of lowering the error floor. Compared to the quantized MSA, the proposed modification is also helpful at low SNRs, where it matches the waterfall performance of the quantized AMSA and OMSA. The new algorithm is designed based on the assumption that trapping/absorbing sets (or other problematic graphical objects) are the major cause of the error floor for quantized LDPC decoders, and it aims to reduce the probability that these problematic objects lead to decoding errors.

Absorbing sets are known to be the primary factor in the error-floor performance of low-density parity-check (LDPC) codes with message passing decoders over the additive white Gaussian noise (AWGN) channel. Besides showing excellent waterfall performance, spatially coupled LDPC (SC-LDPC) codes that are constructed by an edge spreading technique are known to have fewer cycles and absorbing sets than their block code counterparts, and therefore to exhibit better error-floor performance. Based on our previously obtained results for quantized LDPC block decoders, we derive lower bounds on the performance of quantized SC-LDPC decoders, including both a flooding schedule decoder and a sliding window decoder. Numerical simulation results confirm the accuracy of the obtained bounds and show that, for quantized decoders, properly designed SC-LDPC codes have better error-floor performance than their underlying LDPC block codes.

A code-independent performance bound for a given absorbing set is derived for quantized low-density parity-check (LDPC) decoders. The analysis demonstrates that each absorbing set in the Tanner graph imposes a specific lower bound on the frame error rate (FER) of any code containing that absorbing set under a given quantization scheme. This approach is applicable to any message-passing (MP) decoding algorithm and any uniform or non-uniform quantization scheme for LDPC codes. Simulation results using the sum-product algorithm (SPA) provide FERs that are consistent with the obtained bounds. In addition, the bounds demonstrate that the conventional quantized SPA is not capable of achieving very low FERs if the LDPC codes contain certain absorbing sets.