In this work, we develop a reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes over erasures. Our decoder combines classical inactivation decoding, which integrates peeling with symbolic guessing, with a new dual peeling procedure. In the dual peeling stage, we perform row operations on the stabilizer matrix to efficiently reveal stabilizer generators and their linear combinations whose support lies entirely on the erased set. Each such stabilizer identified allows us to freely fix a bit in its support without affecting the logical state of the decoded result. This removes one degree of freedom that would otherwise require a symbolic guess, reducing the number of inactivated variables and decreasing the size of the final linear system that must be solved. We further show that dual peeling combined with standard peeling alone, without inactivation, is sufficient to achieve ML for erasure decoding of surface codes. Simulations across several QLDPC code families confirm that our decoder matches ML logical failure performance while significantly reducing the complexity of inactivation decoding, including more than a 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates.

Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and error patterns with the same syndrome due to code degeneracy. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. This decimation process is applicable to both binary and quaternary BP and it involves sequentially freezing the value of the most reliable qubits to encourage BP convergence. We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations. To explore how and why BPGD improves performance, we discuss several interpretations of BPGD and their connection to BP syndrome decoding.

In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code lengthn, we can compute the weight distribution of anypolar cosetsin timeO(n2). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call themixing factor. To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at lengthn= 128.

Bit-interleaved coded modulation (BICM) and multilevel coded modulation (MLC) are commonly used to combine polar codes with high order modulation. While BICM benefits from simple design and the separation of coding and modulation, MLC shows better performance under successive-cancellation de-coding. In this paper we propose a hybrid polar coded modulation scheme that lies between BICM and MLC, wherein a fraction of bits are assigned to set-partition (SP) labeling and the remaining bits are assigned for Gray labeling. The SP labeled bits undergo sequential demodulation, using iterative demodulation and polar decoding similar to MLC, whereas the Gray labeled bits are first demodulated in parallel and then sent for decoding similar to BICM. Either polar codes or other channel codes (such as LDPC codes) can be used for the Gray labeled bits. For length 2048 rate 1/2 polar code on 256-QAM, the performance gap be-tween BICM (Gray labeling only) and MLC (SP labeling only) can be almost fully closed by the hybrid scheme. Notably, the hybrid scheme has a significant latency advantage over MLC. These performance gains make the proposed scheme attractive for future communication systems such as 6G.

Successive-cancellation list (SCL) decoding is a widely used and studied decoding algorithm for polar codes. For short blocklengths, empirical evidence shows that SCL decoding with moderate list sizes (say,$L \leq 32$) closely matches the performance of maximum-likelihood (ML) decoding. Hashemi et al. proved that on the binary erasure channel (BEC), SCL decoding actually coincides with ML decoding for list sizes$L \geq 2^{\gamma}$, where$\gamma$is a new parameter we call the mixing factor. Loosely speaking, the mixing factor counts the number of information bits mixed-in among the frozen bits; more precisely$\gamma=\vert\{i\in \mathcal{F}^{c}:i\leq \max\{\mathcal{F}\}\}\vert$, where$\mathcal{F}\subset [n]$denotes the set of frozen indices. Herein, we extend the aforementioned result of Hashemi et al. from the BEC to arbitrary binary-input memoryless symmetric channels. Our proof is based on capturing all$2^{\gamma}$decoding paths that correspond to the$\gamma$information bits appearing before the last frozen bit, and then finding the most-likely extension for each of these paths efficiently using a nearest coset decoding algorithm introduced herein. Furthermore, we present a hybrid successive-cancellation list (H -SCL) decoding algorithm, which is a hybrid between conventional SCL decoding and nearest coset decoding. We believe that the hybrid algorithm can outperform the conventional SCL decoder with lower decoding complexity.

The bee-identification problem, formally defined by Tandon, Tan and Varshney (2019), requires the receiver to identify “bees” using a set of unordered noisy measurements. In this previous work, Tandon, Tan and Varshney studied error exponents and showed that decoding the measurements jointly results in a significantly smaller error exponent. Here, we study efficient ways of performing joint decoding. First, by reducing to the problem of finding perfect matching and minimum-cost matchings, we obtain joint decoders that run in time quadratic and cubic in the number of “bees” for the binary erasure (BEC) and binary symmetric channels (BSC), respectively. Next, by studying the matching algorithms in the context of channel coding, we further reduce the running times by using classical tools like peeling decoders and list-decoders. In particular, we show that our identifier algorithms when used with Reed-Muller codes terminates in almost linear and quadratic time for BEC and BSC, respectively.

We present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distributions that arise in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, we can compute the entire weight distribution of certain polar cosets in time$O(n^{2})$. Any polar code can be represented as a disjoint union of such cosets; moreover, this representation extends to polar codes with dynamically frozen bits. This implies that our methods can be also used to compute the weight distribution of polar codes with CRC precoding, of pol-arization-adjusted convolutional (PAC) codes and, in fact, general linear codes. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To reduce the exponential complexity of our algorithm, we make use of the fact that polar codes have a large automorphism group, which includes the lower-triangular affine group LTA$(m,2)$. We prove that LTA$(m,2)$acts transitively on certain subsets of polar codes, thereby drastically reducing the number of polar cosets we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of any polar code of length up to$n=128$.

The problem of channel polarization for an arbitrary sequence$\{W_{i}\}_{i=0}^{n-1}$of$n$independent channels, referred to as a nonstationary sequence of channels, is considered. Also, each of the channels is used only once for communication. We consider a general framework for polarization of non-stationary channels and aim at optimizing the framework toward obtaining the best polarization. This framework includes permuting channels before Arıkan's pairwise channel combining operations are applied at each polarization level and skipping certain combining operations. We define an explicit optimization problem with the objective of finding the best permutation and indices of skipped operations in order to minimize a certain measure of polarization in one-level polarization. We then provide a complete solution to this optimization problem in the case of non-stationary binary erasure channels (BECs). We also propose a greedy method for polarizing non-stationary BECs, based on our solution for one-level polarization. Numerical results confirm the superiority of our method, in terms of various performance metrics, for constructing polar codes in certain non-stationary settings compared to prior work.

Successive-cancellation decoding has gained much renewed interest since the advent of polar coding a decade ago. For polar codes, successive-cancellation decoding can be accomplished in time O(n log n). However, the complexity of successive-cancellation decoding for other families of codes remains largely unexplored. Herein, we prove that successive-cancellation decoding of general binary linear codes is NP-hard. In order to establish this result, we reduce from maximum-likelihood decoding of linear codes, a well-known NP-complete problem. Unlike maximum-likelihood decoding, however, the successive-cancellation decoding problem depends on the choice of a generator matrix. Thus we further strengthen our result by showing that there exist codes for which successive-cancellation decoding remains hard for every possible choice of the generator matrix. On the other hand, we also observe that polynomial-time successive-cancellation decoding can be extended from polar codes to many other linear codes. Finally, we show that every binary linear code can be encoded as a polar code with dynamically frozen bits. This approach makes it possible to use list-decoding of polar codes to approximate the maximum-likelihood decoding performance of arbitrary codes.

The imbalance of a binary word refers to the absolute difference between the number of ones and zeros in the word. Motivated by applications in DNA-based data storage and the success of polar codes, we study the problem of reducing imbalance in the codewords of a polar code. To this end, we adapt the technique of Mazumdar, Roth, and Vontobel by considering balancing sets that correspond to low-order Reed-Muller (RM) codes. Such balancing sets are likely to be included as subcodes in polar codes.Specifically, using the first-order RM code, we show that any message can be encoded into a length-n polar codeword with imbalance at most o(n) in O(nlogn)-time. We then reduce the imbalance even further using two methods. First, we constrain the ambient space $\mathbb{X}$ and analyze the imbalance that the first-order RM code can achieve for words in $\mathbb{X}$. We demonstrate that for codelengths up to 128, the first-order RM code achieves zero imbalance for appropriate choices of $\mathbb{X}$ that sacrifice only a few message bits. Second, we augment the balancing set by considering higher order RM codes. We give a simple recursive upper bound for the guaranteed imbalance of RM codes. We also prove that the second-order RM code $\mathbb{R}\mathbb{M}\left( {2,m} \right)$ balances all even-weight words for m ⩽ 5, while the RM code of order m − 3 balances all even-weight words for m ⩾ 5.

Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short block lengths under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short block lengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has remained the state of the art in polar coding since it was first introduced in 2011. Last year, Arıkan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. Such PAC codes provide another dramatic improvement in performance as compared to CRC-aided list decoding. These codes are based primarily upon the following main ideas: replacing CRC precoding with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. Arıkan's simulation results show that PAC codes, resulting from the combination of these ideas, are quite close to finite-length lower bounds on the performance of any code under ML decoding.One of our main goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, L ≥ 128). List decoding has distinct advantages over sequential decoding in certain scenarios such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance under ML decoding. These results indicate that PAC codes are superior to polar codes and Reed-Muller codes, and suggest that the goal of rate-profiling may be to optimize the weight distribution at low weights.

This paper provides a reduced-complexity approach to maximum likelihood (ML) decoding of cyclic codes. A cyclic code with generator polynomial gcyclic(x) may be considered a terminated convolutional code with a nominal rate of 1. The trellis termination redundancy lowers the rate from 1 to the actual rate of the cyclic code. The proposed decoder represents gcyclic(x) as the product of two polynomials, a convolutional code (CC) polynomial gcc(x) and a cyclic redundancy check (CRC) polynomial gcrc(x), i.e., gcyclic(x) = gcc(x)gcrc(x). This representation facilitates serial list Viterbi algorithm (S-LVA) decoding. Viterbi decoding is performed on the natural trellis for gcc(x), and gcrc(x) is used as a CRC to determine when the S-LVA should conclude. At typical target frame error rates, the expected list size of S-LVA is small, and the average decoding complexity is dominated by the trellis complexity of gcc(x) rather than gcyclic(x). Some high-rate binary Bose-Chaudhuri- Hocquenghem (BCH) examples show that the proposed use of S-LVA via factorization significantly lowers complexity as compared to using the minimum-complexity trellis representation of gcyclic(x) for soft ML decoding.

Polar coding has found its way into many realms in communications and information theory. In most implementation setups, they are accompanied with the list successive cancellation (LSC) decoding algorithm which is shown to provide a superior error performance compared to the original successive cancellation (SC) decoding method. While the SC decoding is fairly well-studied, the exact math behind LSC's superior performance still remains to be of mystery. Multiple techniques have been proposed to further improve the LSC's error performance or to reduce its computational complexity, which are usually motivated by heuristic reasons and shown through numerical simulations. Most notable example is the CRC-aided LSC, which drastically improves the LSC's performance by concatenating the polar code with some high-rate cyclic redundancy check (CRC) codes.In this paper, we present polar codes that are concatenated with an underlying high-rate convolutional code, which are shown to have superior performances over CRC-aided LSC. We also present a computationally-efficient decoding algorithm for these codes which resembles the techniques used in the Viterbi algorithm, and hence is called the convolutional decoding algorithm. To do this, we revisit the error analysis of the original SC decoding along with the concept of Arıkan's helper genie. We address some shortcomings of the CRC-aided LSC and discuss how to turn around them by emulating a convolutional code instead of a CRC code. Contrary to CRC codes, most of the convolutional codes are not a proper choice for concatenation with polar codes. We introduce the bucketing algorithm to construct suitable punctured convolutional codes for this purpose. The proposed framework can accommodate any such underlying convolutional code, which allows one to search for the optimal convolutional code based on their design parameters.

Polar coding gives rise to the first explicit family of codes that provably achieve capacity for a wide range of channels with efficient encoding and decoding. But how fast can polar coding approach capacity as a function of the code length? In finite-length analysis, the scaling between code length and the gap to capacity is usually measured in terms of the scaling exponent μ. It is well known that the optimal scaling exponent, achieved by random binary codes, is μ = 2. It is also well known that the scaling exponent of conventional polar codes on the binary erasure channel (BEC) is μ = 3.627, which falls far short of the optimal value. On the other hand, it was recently shown that polar codes derived from ℓ × ℓ binary polarization kernels approach the optimal scaling exponent μ = 2 on the BEC as ℓ→∞, with high probability over a random choice of the kernel. Herein, we focus on explicit constructions of ℓ×ℓ binary kernels with small scaling exponent for ℓ ≤ 64. In particular, we exhibit a sequence of binary linear codes that approaches capacity on the BEC with quasi-linear complexity and scaling exponent μℓtransforms an underlying BEC into ℓ bit-channels W1, W2>,..., Wℓ. The erasure probabilities of W1, W2>,..., Wℓ, known as the polarization behavior of Kℓ, determine the resulting scaling exponent μ(Kℓ). We first introduce a class of self-dual binary kernels and prove that their polarization behavior satisfies a strong symmetry property. This reduces the problem of constructing Kℓto that of producing a certain nested chain of only ℓ/2 self-orthogonal codes. We use nested cyclic codes, whose distance is as high as possible subject to the orthogonality constraint, to construct the kernels K32and K64. In order to evaluate the polarization behavior of K32and K64, two alternative trellis representations (which may be of independent interest) are proposed. Using the resulting trellises, we show that μ(K32) = 3.122 and explicitly compute over half of the polarization-behavior coefficients for K64, at which point the complexity becomes prohibitive. To complete the computation, we introduce a Monte-Carlo interpolation method, which produces the estimate μ(K64) ≃ 2.87. We augment this estimate with a rigorous proof that μ(K64) <; 2.97.