We analyze polarization-adjusted convolutional codes using the algebraic representation of polar and Reed-Muller codes. We define a large class of codes, called generalized polynomial polar codes which include PAC codes and Reverse PAC codes. We derive structural properties of generalized polynomial polar codes, such as duality, minimum distance. We also deduce some structural limits in terms of number of minimum weight codewords, and dimension of monomial sub-code.

Bivariate bicycle (BB) codes are a prominent class of quantum LDPC codes constructed from group algebras. While the logical dimension and quantum distance of \emph{coprime} BB codes are known to be determined by a greatest common divisor polynomial $g(z)$, the properties governing their fault tolerance under noisy measurement have remained implicit. In this work, we prove that this same polynomial $g(z)$ dictates the code's stabilizer redundancy and the structure of the classical \emph{syndrome codes} required for single-shot decoding. We derive a strict equality between the quantum rate and the stabilizer redundancy density, and we provide BCH-like bounds on the achievable single-shot measurement error tolerance. Guided by this framework, we construct small coprime BB codes with significantly improved syndrome distance ($d_S$) and evaluate them using BP+OSD. Our analysis reveals a structural bottleneck: within the coprime BB ansatz, high quantum rate imposes an upper bound on syndrome distance, limiting single-shot performance. These results provide concrete algebraic design rules for next-generation 2BGA codes in measurement-limited architectures.

Bias-tailored codes such as the XZZX surface code and the domain wall color code achieve high dephasing-biased thresholds because, in the infinite-bias limit, their $Z$ syndromes decouple into one-dimensional repetition-like chains; the $X^3Z^3$ Floquet code shows an analogous strip-wise structure for detector events in spacetime. We capture this common mechanism by defining strip-symmetric biased codes, a class of static stabilizer and dynamical (Floquet) codes for which, under pure dephasing and perfect measurements, each elementary $Z$ fault is confined to a strip and the Z-detector--fault incidence matrix is block diagonal. For such codes the Z-detector hypergraph decomposes into independent strip components and maximum-likelihood $Z$ decoding factorizes across strips, yielding complexity savings for matching-based decoders. We characterize strip symmetry via per-strip stabilizer products, viewed as a $\mathbb{Z}_2$ 1-form symmetry, place XZZX, the domain wall color code, and $X^3Z^3$ in this framework, and introduce synthetic strip-symmetric detector models and domain-wise Clifford constructions that serve as design tools for new bias-tailored Floquet codes.

Polar codes can be viewed as decreasing monomial codes, revealing a rich algebraic structure governed by the lower-triangular affine (LTA) group. We develop a general framework to compute the Hamming weight of codewords generated by sums of monomials, express these weights in a canonical dyadic form, and derive closed expressions for key structural templates (disjoint sums, nested blocks, complementary flips) that generate the low and intermediate weight spectrum. Combining these templates with the LTA group action, we obtain explicit multiplicity formulas, yielding a unified algebraic method to characterize and enumerate codewords.

In this article, we provide a characterization of Type-I codewords with weight less than twice the minimum distance of polar codes, based on the action of the lower triangular affine group. We present a closed-form formula for the enumeration of such codewords. In addition, we propose an improved weight contribution partial order.

Polar codes are constructed based on the reliability of sub-channels resulting from the polarization effect. However, this information-theoretic construction approach leads to a poor weight distribution. To address this issue, pre-transformed polar codes, such as CRC-polar codes and PAC codes, have been employed. In this paper, we focus on the structure of polar codes without applying any pre-transformations and explore methods, guided by the weight-contribution partial order, to design polarlike codes with enhanced weight distribution, notably without employing any search or optimization algorithms. Numerical results demonstrate improvement over a range of codes both with and without pre-transformation.

This paper analyzes the formation of minimum-weight codewords (MWCs) in cyclic redundancy check polar (CRC-polar) codes, known for their competitive performance, and demonstrates how the number of MWCs significantly reduces relative to polar codes, which is the reason for the superiority of CRC-polar codes. Inspired by the reason behind the significant reduction of MWCs, we propose a modified reliability-based rate-profile for polarization-adjusted convolutional (PAC) codes, termed Profile-Shifted PAC (PS-PAC) codes, designed to achieve a similar reduction in MWCs through precoding. The results demonstrate a significant improvement in error performance compared to CRC-polar codes, achieving up to a 0.5 dB power gain with short PS-PAC codes. Furthermore, we enhance CRC-Polar codes by leveraging convolutional precoding in PAC codes to implement a continuous deployment (masking) of parity check bits over frozen bits, called Continuous CRC-Polar codes. This approach enhances performance for medium-length codes, with an overall improvement of 0.12 dB.

The ordered-reliability bits (ORB) variant of guessing random additive noise decoding (GRAND), known as ORBGRAND, achieves remarkably low time complexity at high code rates compared to other GRAND variants. However, its computational complexity remains higher than other near-ML universal decoders like ordered-statistics decoding (OSD). To address this, we propose segmented ORBGRAND, which partitions the error pattern search space based on code properties, generates syndrome-consistent sub-patterns (reducing invalid error patterns), and combines them in a near-ML order using sub-weights derived from two-level integer partitions of logistic weight. Numerical results show that segmented ORBGRAND reduces the average number of queries by at least 66% across all SNRs and cuts basic operations by over an order of magnitude, depending on segmentation and code rate. Further efficiency gains come from leveraging pre-generated shared sub-patterns, reducing average decoding time. Additionally, with abandonment ($b = 10^{5}$or smaller), segmented ORBGRAND provides a 0.2 dB power gain over ORBGRAND.

The structure of a linear block code is pivotal in defining fundamental properties, particularly weight distribution, and code design. In this study, we characterize the Type II structure of polar codewords with weights less than twice the minimum weight wmin, using the lower triangular affine (LTA) transform. We present a closed-form formula for their enumeration. Leveraging this structure and additionally characterizing the structure of weight 2wmin, we ascertain the complete weight distribution of low-rate polar codes (with minimum distance dmin= 2m−2where code-length is 2m) and, through the utilization of dual codes properties, high-rate polar codes, subcodes of Reed–Muller (RM) codes, and RM–Polar codes. Furthermore, we introduce a new partial order based on the weight distribution and explore its properties and applications in code construction and analysis.

Convolutional precoding in polarization-adjusted convolutional (PAC) codes is a recently introduced variant of polar codes. It has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction has the potential to significantly improve the error correction performance. From a codeword formation perspective, this reduction has limitations. Capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper studies reverse precoding that can effectively reduce minimum-weight codewords more than in PAC codes. We propose a look-ahead list decoding for the reverse PAC codes, which has the same order of complexity as list decoding in PAC codes. Through numerical analysis, we demonstrate a notable reduction in error coefficients compared to PAC codes and polar codes, resulting in a remarkable improvement in the block error rate, in particular at high code rates.

The structure of a linear block code is pivotal in defining fundamental properties, particularly weight distribution, and code design. In this study, we characterize the Type II structure of polar codewords with weights less than twice the minimum weight Wmin, utilizing the lower triangular affine (LTA) transform. We present a closed-form formula for their enumer-ation. Leveraging this structure and additionally characterizing the structure of weight 2 Wmin, we ascertain the complete weight distribution of low-rate and, through the utilization of dual codes properties, high-rate polar codes, subcodes of Reed-Muller (RM) codes, and RMxPolar codes. Furthermore, we introduce a partial order based on the weight distribution and briefly explore its properties and applications in code construction and analysis.

The weight distribution of an error correction code is a critical determinant of its error-correcting performance. In the case of polar codes, the minimum weight wmin(equal to the minimum distanced) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight 1.5wmin. The key insight for code design is that the enumeration of codewords with weight wmin and 1.5wminrelies on the set of maximum degree monomials. This set corresponds to the indices of minimum weight rows of the polar transformGNbelonging to the information setI. Consequently, reducing the cardinality of this set can lead to a reduction of the number of codewords in both weight categories.

Convolutional precoding in polarization-adjusted convolutional (PAC) codes, a recently introduced variant of polar codes, has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction has the potential to significantly improve the error correction performance. From a codeword formation perspective in cosets, this reduction has a limitation in the PAC coding which depends on the rows of the generator matrix involved in the formation of codewords. To overcome this limitation, capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper introduces a novel precoding scheme that strategically disrupts the formation of a majority of minimum-weight codewords. This scheme significantly enhances the error coefficient without compromising the minimum distance of the code. Through numerical analysis, we demonstrate a noteworthy reduction in error coefficients compared to PAC codes and polar codes, resulting in a remarkable improvement in the block error rate of short codes.

The recently introduced maximum-likelihood (ML) decoding scheme called guessing random additive noise decoding (GRAND) has demonstrated a remarkably low time complexity in high signal-to-noise ratio (SNR) regimes. However, the complexity is not as low at low SNR regimes and low code rates. To mitigate this concern, we propose a scheme for a near-ML variant of GRAND called ordered reliability bits GRAND (or ORBGRAND), which divides codewords into segments based on the properties of the underlying code, generates sub-patterns for each segment consistent with the syndrome (thus reducing the number of inconsistent error patterns generated), and combines them in a near-ML order using two-level integer partitions of logistic weight. The numerical evaluation demonstrates that the proposed scheme, called segmented ORBGRAND, reduces the average number of queries (time complexity/latency) to one-third at all SNR regimes. Moreover, the segmented ORBGRAND with abandonment also improves the error correction performance.

Minimum weight codewords play a crucial role in the error correction performance of a linear block code. In this work, we establish an explicit construction for these codewords of polar codes as a sum of the generator matrix rows, which can then be used as a foundation for two applications. In the first application, we obtain a lower bound for the number of minimum-weight codewords (a.k.a. the error coefficient), which matches the exact number established previously in the literature. In the second application, we derive a novel method that modifies the information set (a.k.a. rate profile) of polar codes and PAC codes in order to reduce the error coefficient, hence improving their performance. More specifically, by analyzing the structure of minimum-weight codewords of polar codes (as special sums of the rows in the polar transform matrix), we can identify rows (corresponding to information bits) that contribute the most to the formation of such codewords and then replace them with other rows (corresponding to frozen bits) that bring in few minimum-weight codewords. A similar process can also be applied to PAC codes. Our approach deviates from the traditional constructions of polar codes, which mostly focus on the reliability of the sub-channels, by taking into account another important factor - the weight distribution. Extensive numerical results show that the modified codes outperform PAC codes and CRC-Polar codes at the practical block error rate of$10^{-2}$-$10^{-3}$.

Maximum-likelihood (ML) decoding can be used to obtain the optimal performance of error correction codes. However, the size of the search space and consequently the decoding complexity grows exponentially, making it impractical to be employed for long codes. In this paper, we propose an approach to constrain the search space for error patterns under a recently introduced near ML decoding scheme called guessing random additive noise decoding (GRAND). In this approach, the syndrome-based constraints which divide the search space into disjoint sets are progressively evaluated. By employing p constraints extracted from the parity check matrix, the average number of queries reduces by a factor of 2pwhile the error correction performance remains intact.

Polar codes are normally constructed based on the reliability of the sub-channels in the polarized vector channel. Code construction based on reliability is compatible with successive cancellation decoding. However, due to poor Hamming distance properties, the designed codes cannot perform well with near maximum likelihood decoders. In this work, we propose a new approach that modifies polar codes and PAC codes to significantly lower the number of codewords with minimum distance (a.k.a. error coefficient). This approach is based on the recognition of all the rows of polar transform involved in the formation of the minimum-weight codewords. The numerical results show that the designed codes outperform polar codes and PAC codes under list decoding.

The number of minimum weight codewords, a.k.a error coefficient, is a good comparative measure for the block error rate (BLER) of a linear block code, in particular in a high SNR regime. The smaller the error coefficient, the lower the BLER. Unlike polar codes, the error coefficient of polarization-adjusted convolutional (PAC) codes cannot be determined easily due to the precoding stage. In this work, we propose an enumeration method that considers the impact of convolutional precoding on the minimum weight codewords of polar codes. This relative enumeration method simplifies the process significantly compared with conventional methods as the complexity analysis shows.

The successive cancellation (SC) process in which symbols are decoded sequentially by processing some intermediate information is an essential component of various decoding algorithms used for polar codes and their variants. In some decoding schemes, we may need to redo this process from some specific symbol or from the first symbol. This operation is called rewinding. Rewinding the SC process seems trivial if we have access to all intermediate log-likelihood ratios (LLRs) and partial sums. However, as the block length increases, retaining all of the intermediate information becomes inefficient and impractical. Rewinding the SC process in a memory-efficient way is a problem that we address in this paper. As we store a fraction of all the intermediate information in the memory-efficient scheme, we may not be able to rewind the SC process to the target symbol index. The reason is that some of the stored intermediate information needed to decode the target symbol may have been overwritten. To recompute the lost information, we may need to rewind the process further. Before proposing the formal scheme for the rewinding process, we explore the known properties of the SC process based on the binary representation of the bit indices. Then, we introduce a new operator used for grouping the bit indices. This special grouping helps us in finding the closest bit index to the target index for rewinding. We also analytically prove that this approach gives access to the untouched intermediate information stored in the memory which is essential in resuming the SC process. Finally, we adapt the proposed approach to multiple rewinds and apply it to SC-flip decoding and shifted-pruning-based list decoding. The numerical evaluation of the proposed solution shows a significant reduction of ≥50% in the complexity of the additional decoding attempts at medium and high SNR regimes for SC-flip decoding and less for shifted-pruning based list decoding.

Polar codes are constructed based on the reliability of bit-channels. This construction suits the successive cancellation (SC) decoding, where one error in the successive estimation of the bits fails the decoding. However, in SC list (SCL) decoding, the correct path may remain in the list by tolerating multiple penalties. This characteristic of list decoding demands a different approach in code construction. In this work, we modify the conventional construction by a greedy search algorithm in which a bit-swapping approach is employed to re-distribute the low-reliability bits in the subblocks aiming for a reduction in the probability of correct path elimination. The numerical results for polar codes of length 1 kb under CRC-aided SCL decoding show improvements of about 0.4 dB for R=0.8 and over 0.2 dB for R=0.5 at L=32.

In successive cancellation list (SCL) decoding, the list pruning operation retains the L paths with highest likelihoods. However, the correct path might be among the paths with low likelihoods due to channel noise. In this case, the correct path is eliminated from the list. In this work, we study the event of elimination of the correct path and we analyze where and how this event occurs. A modified pruning scheme named shifted-pruning over a set of low-reliability bit-channels named critical bits is proposed aiming to avoid the elimination of the correct path in additional decoding attempts after a decoding failure occurs. Shifted-pruning is realized by selecting the paths k + 1 to k + L out of the 2L ordered paths instead of the paths 1 to L. The numerical results for polar codes of length 512 and code rates 0.5 and 0.8 and list sizes L = 2, 8 and 32 show that the shifted-pruning scheme is a low-complexity equivalent to the bit-flipping scheme while it can outperform the bit-flipping method by providing 0. 25-0.5dB gain.