arXiv subjects

Mohammad Rowshan

Publications and source records attributed to Mohammad Rowshan.

At least 19 recordsLinked to original sources

arXiv2026

Structural Analysis of Directional qLDPC Codes

Directional codes, recently introduced by Geh\'er--Byfield--Ruban \cite{Geher2025Directional}, constitute a hardware-motivated family of quantum low-density parity-check (qLDPC) codes. These codes are defined by stabilizers measured by ancilla qubits executing a fixed \emph{direction word} (route) on square- or hex-grid connectivity. In this work, we develop a comprehensive \emph{word-first} analysis framework for route-generated, translation-invariant CSS codes on rectangular tori. Under this framework, a direction word $W$ deterministically induces a finite support pattern $P(W)$, from which we analytically derive: (i)~a closed-form route-to-support map; (ii)~the odd-multiplicity difference lattice $L(W)$ that classifies commutation-compatible $X/Z$ layouts; and (iii)~conservative finite-torus admissibility criteria. Furthermore, we provide: (iv)~a rigorous word equivalence and canonicalization theory (incorporating dihedral lattice symmetries, reversal/inversion, and cyclic shifts) to enable symmetry-quotiented searches; (v)~an ``inverse problem'' criterion to determine when a translation-invariant support pattern is realizable by a single route, including reconstruction and non-realizability certificates; and (vi)~a quasi-cyclic (group-algebra) reduction for row-periodic layouts that explains the sensitivity of code dimension $k$ to boundary conditions. As a case study, we analyze the word $W=\texttt{NE$^2$NE$^2$N}$ end-to-end. We provide explicit stabilizer dependencies, commuting-operator motifs, and an exact criterion for dimension collapse on thin rectangles: for $(L_x, L_y) = (2d, d)$ with row alternation, we find $k=4$ if $6 \mid d$, and $k=0$ otherwise.

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arXiv2026

Native Non-Clifford Gates in Quantum LDPC Codes: Conditions, Synthesis, and Scaling Limits

Native constant-depth non-Clifford gates on quantum low-density parity-check (qLDPC) codes can substantially reduce the space-time overhead of magic-state distillation. This paper investigates the underlying parity and structural conditions governing parallel non-Clifford gate implementations. By analyzing their invariance, algebraic forms, and circuit synthesis, we establish fundamental limits on code scaling: strict saturated implementations cannot achieve constant-depth realization as code distance grows due to tight distance-depth bounds. We evaluate alternative scaling routes, showing that scalable designs must either relax strict subspace requirements or manage gate congestion. Through analytical bounds and code searches, we demonstrate that reconciling parallel non-Clifford operations with linear distance requires navigating these fundamental structural trade-offs.

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arXiv2026

Algebraic Properties of PAC Codes

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.

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arXiv2026

Generalized Weight Structure of Polar Codes: Selected Template Polynomials

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.

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arXiv2026

A Hybrid Reliability--Weight Framework for Construction of Polar Codes

Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices.

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arXiv2026

Bias-Aware BP Decoding of Quantum Codes via Directional Degeneracy

We study directionally informed belief propagation (BP) decoding for quantum CSS codes, where anisotropic Tanner-graph structure and biased noise concentrate degeneracy along preferred directions. We formalize this by placing orientation weights on Tanner-graph edges, aggregating them into per-qubit directional weights, and defining a \emph{directional degeneracy enumerator} that summarizes how degeneracy concentrates along those directions. A single bias parameter~$\beta$ maps these weights into site-dependent log-likelihood ratios (LLRs), yielding anisotropic priors that plug directly into standard BP$\rightarrow$OSD decoders without changing the code construction. We derive bounds relating directional and Hamming distances, upper bound the number of degenerate error classes per syndrome as a function of distance, rate, and directional bias, and give a MacWilliams-type expression for the directional enumerator. Finite-length simulations under code-capacity noise show significant logical error-rate reductions -- often an order of magnitude at moderate physical error rates -- confirming that modest anisotropy is a simple and effective route to hardware-aware decoding gains.

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arXiv2026

Strip-Symmetric Quantum Codes for Biased Noise: Z-Decoupling in Stabilizer and Floquet Codes

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.

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arXiv2026

Single-Shot and Few-Shot Decoding via Stabilizer Redundancy in Bivariate Bicycle Codes

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.

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arXiv2025

Towards Weight Distribution-Aware Polar Codes

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 polar-like 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.

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arXiv2025

On Weight Enumeration and Structure Characterization of Polar Codes via Group Actions

In this article, we provide a complete characterization of codewords in polar codes with weights less than twice the minimum distance, using the group action of the lower triangular affine (LTA) group. We derive a closed-form formula for the enumeration of such codewords. Furthermore, we introduce an enhanced partial order based on weight contributions, offering refined tools for code design. Our results extend previous work on Type II codewords to a full description of Type I codewords and offer new insights into the algebraic structure underlying decreasing monomial codes, including polar and Reed-Muller codes.

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arXiv2025

PAC Codes Meet CRC-Polar Codes

CRC-Polar codes under SC list decoding are well-regarded for their competitive error performance. This paper examines these codes by focusing on minimum weight codewords, breaking them down into the rows of the polar transform. Inspired by the significant impact of parity check bits and their positions, we apply a shifted rate-profile for polarization-adjusted convolutional (PS-PAC) codes, thereby achieving similar improvements in the weight distribution of polar codes through precoding. The results demonstrate a significant improvement in error performance, achieving up to a 0.5 dB power gain with short PS-PAC codes. Additionally, leveraging convolutional precoding in PAC codes, we adopt a continuous deployment (masking) of parity check bits derived from the remainder of continuous division of the partial message polynomial and the CRC polynomial over frozen positions in the rate-profile. This approach enhances performance for medium-length codes, with an overall improvement of 0.12 dB.

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arXiv2024

Reverse PAC Codes: Look-ahead List Decoding

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.

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arXiv2024

Channel Coding Toward 6G: Technical Overview and Outlook

Channel coding plays a pivotal role in ensuring reliable communication over wireless channels. With the growing need for ultra-reliable communication in emerging wireless use cases, the significance of channel coding has amplified. Furthermore, minimizing decoding latency is crucial for critical-mission applications, while optimizing energy efficiency is paramount for mobile and the Internet of Things (IoT) communications. As the fifth generation (5G) of mobile communications is currently in operation and 5G-advanced is on the horizon, the objective of this paper is to assess prominent channel coding schemes in the context of recent advancements and the anticipated requirements for the sixth generation (6G). In this paper, after considering the potential impact of channel coding on key performance indicators (KPIs) of wireless networks, we review the evolution of mobile communication standards and the organizations involved in the standardization, from the first generation (1G) to the current 5G, highlighting the technologies integral to achieving targeted KPIs such as reliability, data rate, latency, energy efficiency, spectral efficiency, connection density, and traffic capacity. Following this, we delve into the anticipated requirements for potential use cases in 6G. The subsequent sections of the paper focus on a comprehensive review of three primary coding schemes utilized in past generations and their recent advancements: low-density parity-check (LDPC) codes, turbo codes (including convolutional codes), polar codes (alongside Reed-Muller codes). Additionally, we examine alternative coding schemes like Fountain codes and sparse regression codes. Our evaluation includes a comparative analysis of error correction performance and the performance of hardware implementation for these coding schemes, providing insights into their potential and suitability for the upcoming 6G era.

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arXiv2024

Weight Structure of Low/High-Rate Polar Codes and Its Applications

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 $w_{min}$, utilizing 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 $2w_{min}$, 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.

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arXiv2023

Segmented GRAND: Complexity Reduction through Sub-Pattern Combination

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. Furthermore, with abandonment ($b=10^{5}$ or smaller), segmented ORBGRAND provides a 0.2 dB power gain over ORBGRAND. Additionally, we provide an analytical justification for why the logistic weight-based ordering of error patterns in ORBGRAND closely approximates the ML order and discuss the underlying assumptions of ORBGRAND.

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arXiv2023

On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords

The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum distance $d$) 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 $1.5\wm$. Our simulations demonstrate the potential for extending this method to higher weights.

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arXiv2022

Selective Reverse PAC Coding for Sphere Decoding

Convolutional precoding in polarization-adjusted convolutional (PAC) codes can reduce the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This can result in improving the error correction performance of (near) maximum likelihood (ML) decoders such as sequential decoders and sphere decoders. However, PAC codes cannot be decoded by sphere decoding. The reason is twofold: 1) Sphere decoding of polar codes is performed from the last bit - due to the lower rectangular shape of the polar transform. Whereas the shape of PAC codes generator matrix is no longer triangular. 2) One may modify the precoding matrix to get a lower-triangular shape. However, this may reduce the minimum distance of the code due to the formation of unwanted cosets. This work proposes a selective convolutional precoding scheme with transposed precoding matrix to reduce the error coefficient while avoiding the reduction in the minimum distance. The numerical results show the improvement of block error rate by 0.2-0.6 dB, depending on the code rate, in medium and high SNR regimes.

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arXiv2022

Constrained Error Pattern Generation for GRAND

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 $2^p$ while the error correction performance remains intact.

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