Distance Of Codes Research Articles

Spike and Slab Regression for Nonstationary Gaussian Linear Mixed Effects Modeling of Rapid Disease Progression

ABSTRACTSelect measures of social and environmental determinants of health (referred to as “geomarkers”), predict rapid lung function decline in cystic fibrosis (CF), defined as a prolonged decline relative to patient and/or center‐level norms. The extent to which hyper‐localization, defined as increasing the spatiotemporal precision of geomarkers, aids in prediction of rapid lung decline remains unclear. Linear mixed effects (LME) models with specialized covariance functions have been used for predicting rapid lung function decline, but there are few options to properly incorporate spatial correlation into the covariance functions while inducing simultaneous variable selection. Our innovative Bayesian model uses a spike and slab prior for simultaneous variable selection and offers additional advantages when coupled with nonstationary Gaussian LME modeling. This model also incorporates spatial correlation through an additional random effect term that accounts for spatial correlation based on ZIP code distances. We validated the model with simulations and applied it to real CF data from a Midwestern CF Center. We demonstrate how a combination of demographic, clinical, and geomarker variables can be selected as optimal predictors using Bayesian false discovery rate controlling rule. Our results indicate that incorporating spatiotemporal effects and geomarkers into this novel Bayesian stochastic LME model enhances the dynamic prediction of rapid CF disease progression.

Optimal execution of logical Hadamard with low‐space overhead in rotated surface code

AbstractFault‐tolerant quantum computation requires error‐correcting codes that enable reliable universal quantum operations. This study introduces a novel approach that executes the logical Hadamard with low‐space requirements while preserving the original definition of logical operators within the framework of the rotated surface codes. Our method leverages a boundary deformation method to rotate the logical qubit transformed by transversal Hadamard. Following this, the original encoding of the logical qubit is reinstated through logical flip‐and‐shift operations. The estimated space–time cost for a logical Hadamard operation with a code distance d is 5d2 + 3d2. The efficiency enhancement of the proposed method is approximately four times greater than those of previous approaches, regardless of the code distance. Unlike the traditional method, implementing a logical Hadamard requires only two patches instead of seven. Furthermore, the proposed method ensures the parallelism of quantum circuits by preventing interferences between adjacent logical data qubits.

Access to National Cancer Institute-Designated Cancer Centers Among Native American Cancer Patients

Introduction/Purpose Native Americans (NAs) are subject to high cancer mortality rates in the USA. Despite that, they face significant geographic barriers to access to cancer care. This study aims to estimate the travel distance to a National Cancer Institute (NCI)-designated cancer center for NA patients in Utah and the continental USA. Methods This IRB-approved study utilized retrospective data on genitourinary cancer patients from both NAs and white populations from February 2013 to January 2023. The distance of their geographical location to the Huntsman Cancer Institute (HCI) at the University of Utah was calculated using their home zip code and a GeoData ZIP Code Distance Calculations Matrix Template. A shapefile containing NCI-designated cancer centers was used alongside the Area Deprivation Index (ADI), matched to block groups from the 2020 census, to serve as a national control group. All geographic data was visualized in ArcGIS 10.7 by using the coordinates and a 5-digit zip code tabulation area to map locations. Results A total of 468 NA patients were eligible and included. The median travel distance for NA patients vs. white patients to HCI was 190.6 miles (range: 1.1-596.4 miles) vs. 21.6 miles (range: 1.1-269 miles, p<0.0001). In the continental US, the median travel distance from NA reservations vs. ADI-matched block groups to the nearest NCI-designated cancer centers was 186.5 miles (range 77.8-629 miles) vs. 159 miles (range 1.9-671.3 miles, p<0.01). Conclusion The travel distance to NCI-designated cancer center for NA cancer patients in Utah was around nine times longer than that of white cancer patients. This study highlights the significant disparity in cancer care accessibility faced by NA communities.

Weight-Reduced Stabilizer Codes with Lower Overhead

Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024

Empirical validation of the use of projective techniques in psychological testing using Galois fields

It is shown that the problem of the adequacy of psychological testing methods, which are varieties of “projective techniques”, is far from being universally recognized. To solve this problem, we used an empirical method based on collecting of statistics of respondents’ answers, as well as a method of analyzing this statistics by means of representing permutations through functions taking values in Galois fields. Based on experimental data, it is shown that the distribution of respondents’ answers to a test in which they are asked to rank pictures in accordance with their own preferences is not homogeneous. Experimental data show that there are answer options that are statistically most common. An interpretation of testing is proposed in which passing the test is considered as “connecting” an external additional layer to the neural network formed by the respondent’s brain. In accordance with this interpretation, the most frequently occurring answer options can be considered as the basis for the formation of classification characteristics. It is shown that during using tests of this type it is advisable to take into account the code distances between the answer of a particular respondent and the codes corresponding to the most frequently occurring sequences. The possibilities of generating psychological tests directly based on experimental data and images generated by neural networks are discussed.

Low-power coding method in data transmission systems

The object of the study is the Network-on-Chip (NoC) technology, which has become a popular choice for the on-chip communication architecture of modern System-on-Chip (SoC) devices. The subject matter of the article is methods of reducing dissipated power in NoC and SoC. The goal of the work is: development of a low-power coding method that allows for the efficient transmission or storage of information. The following tasks are solved in the article: analysis of classification methods for combinatorial structures, construction a system of typical representatives and analysis of their characteristics. The research methods are based on the use of set theory, system theory and combinatorics. The following results are obtained: analyzed factors that affect the dissipated power, considered principles of constructing energy-efficient codes. It is shown that switching activity significantly affects the total power and one of the effective methods for reducing switching activity during communication between devices or on-chip communication is the use of low-power coding methods. A method of hierarchical classification of unit distance codes and algorithms for solving step-by-step problems have been developed. The method is based on the invariant approach and construction of a system of different representatives. Estimates of their number have been obtained, characteristics have been determined, and catalogs of typical representatives have been formed. Conclusions. The article analyzes factors that affect dissipated power, and considers the principles of constructing energy-efficient codes. A method of hierarchical classification of single distance codes and algorithms for solving step-by-step problems have been developed, and catalogs of typical representatives have been formed. The application of the developed method will allow developers to analyze and select codes with the best properties and, as a result, obtain better results in terms of network delays, energy costs, and other design limitations for computer systems.

Safeguarding Oscillators and Qudits with Distributed Two-Mode Squeezing

Recent advancements in multi-mode Gottesman-Kitaev-Preskill (GKP) codes have shown great promise in enhancing the protection of both discrete and analog quantum information. This broadened range of protection brings opportunities beyond quantum computing to benefit quantum sensing by safeguarding squeezing — the essential resource in many quantum metrology protocols. However, the potential for quantum sensing to benefit quantum error correction has been less explored. In this work, we provide a unique example where techniques from quantum sensing can be applied to improve multi-mode GKP codes. Inspired by distributed quantum sensing, we propose the distributed two-mode squeezing (dtms) GKP codes that offer benefits in error correction with minimal active encoding operations. Indeed, the proposed codes rely on a single (active) two-mode squeezing element and an array of beamsplitters that effectively distributes continuous-variable correlations to many GKP ancillae, similar to continuous-variable distributed quantum sensing. Despite this simple construction, the code distance achievable with dtms-GKP qubit codes is comparable to previous results obtained through brute-force numerical search \cite{lin2023closest}. Moreover, these codes enable analog noise suppression beyond that of the best-known two-mode codes \cite{noh2020o2o} without requiring an additional squeezer. We also provide a simple two-stage decoder for the proposed codes, which appears near-optimal for the case of two modes and permits analytical evaluation.

Improving Threshold for Fault-Tolerant Color-Code Quantum Computing by Flagged Weight Optimization

Color codes are promising quantum error-correction (QEC) codes because they have an advantage over surface codes in that all Clifford gates can be implemented transversally. However, the thresholds of color codes under circuit-level noise are relatively low, mainly because measurements of their high-weight stabilizer generators cause an increase in the circuit depth and, thus, substantial errors are introduced. This makes color codes not the best candidate for fault-tolerant quantum computing. Here, we propose a method to suppress the impact of such errors by optimizing weights of decoders using conditional error probabilities conditioned on the measurement outcomes of flag qubits. In numerical simulations, we improve the threshold of the (4.8.8) color code under circuit-level noise from 0.14% to around 0.27%, which is calculated by using an integer programming decoder. Furthermore, in the (6.6.6) color code, we achieve a circuit-level threshold of around 0.36%, which is almost the same value as the highest value in the previous studies employing the same noise model. In both cases, the effective code distance is also improved compared to a conventional method that uses a single ancilla qubit for each stabilizer measurement. Thereby, the achieved logical error rates at low physical error rates are almost one order of magnitude lower than those of the conventional method with the same code distance. Even when compared to the single-ancilla method with a higher code distance, considering the increased number of qubits used in our method, we achieve lower logical error rates in most cases. This method can also be applied to other weight-based decoders, making the color codes more promising as candidates for the experimental implementation of QEC. Furthermore, one can utilize this approach to improve a threshold of wider classes of QEC codes, such as high-rate quantum low-density parity-check codes. Published by the American Physical Society 2024

New constructions of MSRD codes

In this work, we provide four methods for constructing new maximum sum-rank distance (MSRD) codes. The first method, a variant of cartesian products, allows faster decoding than known MSRD codes of the same parameters. The other three methods allow us to extend or modify existing MSRD codes in order to obtain new explicit MSRD codes for sets of matrix sizes (numbers of rows and columns in different blocks) that were not attainable by previous constructions. In this way, we show that MSRD codes exist (by giving explicit constructions) for new ranges of parameters, in particular with different numbers of rows and columns at different positions.

An Efficient Construction Method Based on Partial Distance of Polar Codes With Reed-Solomon Kernel

An Efficient Construction Method Based on Partial Distance of Polar Codes With Reed-Solomon Kernel

Approximate Quantum Codes From Long Wormholes

We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, N, of fermions goes to infinity. For SYK, the distance scales as N1/2, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. N.99, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size N. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity.

Two classes of LCD BCH codes over finite fields

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths qm+1q+1 and qm+1 over the finite field Fq, both of which are LCD codes. The dimensions of narrow-sense BCH codes of length qm+1q+1 with designed distance δ=ℓqm−12+1 are determined, where q>2 and 2≤ℓ≤q−1. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length qm+1 are developed for odd q, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length qm+1 being dually-BCH codes are presented, where q is odd and m≢0(mod4).

Classical product code constructions for quantum Calderbank-Shor-Steane codes

Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3-fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

Quantum Lego Expansion Pack: Enumerators from Tensor Networks

We provide the first tensor-network method for computing quantum weight enumerator polynomials in the most general form. If a quantum code has a known tensor-network construction of its encoding map, our method is far more efficient, and in some cases exponentially faster than the existing approach. As a corollary, it produces decoders and an algorithm that computes the code distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can be substantially faster compared to the current methods. We also introduce novel weight enumerators and their applications. In particular, we show that these enumerators can be used to compute logical error rates exactly and thus construct (optimal) decoders for any independent and identically distributed single qubit or qudit error channels. The enumerators also provide a more efficient method for computing nonstabilizerness in quantum many-body states. As the power for these speedups rely on a quantum Lego decomposition of quantum codes, we further provide systematic methods for decomposing quantum codes and graph states into a modular construction for which our technique applies. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the two-dimensional Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force. Published by the American Physical Society 2024

Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma

The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli \cite{AKS06}, we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko \cite{KP13}, we show that any subsystem stabilizer code can be "doubled" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis \cite{BTL10}. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.

New Lower Bounds for the Minimum Distance of Cyclic Codes and Applications to Locally Repairable Codes

New Lower Bounds for the Minimum Distance of Cyclic Codes and Applications to Locally Repairable Codes

Quantum error-correction using humming sparrow optimization based self-adaptive deep cnn noise correction module

The error correction model’s main purpose in heavy hexagonal quantum codes is to improve their reliability for quantum computing applications. Existing challenges include finding the optimal decoder for quantum error correction in heavy hexagonal codes. This research propels the frontier of quantum error correction, with a specific focus on tailoring topological quantum error-correcting codes for the unique challenges posed by superconducting qubits in quantum computers. In response, this research harnesses the power of deep learning, presenting a Humming sparrow optimization based self-adaptive deep CNN (HSO-based SADCNN) model designed for heavy hexagonal codes. This decoder incorporates a Self-adaptive Deep CNN (SADCNN) Noise Correction Module, a sophisticated component to refine error correction. The proposed decoder’s efficacy is rigorously evaluated across varying code distances (three, five, and seven) using the Humming Sparrow Optimization (HSO) algorithm. HSO, intricately designed to fine-tune the SADCNN decoder, significantly enhances its error correction capabilities for heavy hexagonal quantum codes. The algorithm seamlessly integrates advantageous characteristics of herding and tracing from Humming Bird optimization and Sparrow search optimization, representing a critical stride in advancing the reliability of quantum computing applications, particularly within the intricate domain of heavy hexagonal quantum codes. Based upon the achievements, the Training Percentage (TP) 90 metrics demonstrate significant progress, boasting a commendable accuracy of 97.35%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$97.35\\%$$\\end{document}, coupled with reduced logical error probability and a diminished bit error rate, marked at 5.51 and 3.72, respectively.

Distance and grid-like codes support the navigation of abstract social space in the human brain.

People form impressions about others during daily social encounters and infer personality traits from others' behaviors. Such trait inference is thought to rely on two universal dimensions: competence and warmth. These two dimensions can be used to construct a 'social cognitive map' organizing massive information obtained from social encounters efficiently. Originating from spatial cognition, the neural codes supporting the representation and navigation of spatial cognitive maps have been widely studied. Recent studies suggest similar neural mechanism subserves the map-like architecture in social cognition as well. Here we investigated how spatial codes operate beyond the physical environment and support the representation and navigation of social cognitive map. We designed a social value space defined by two dimensions of competence and warmth. Behaviorally, participants were able to navigate to a learned location from random starting locations in this abstract social space. At the neural level, we identified the representation of distance in the precuneus, fusiform gyrus, and middle occipital gyrus. We also found partial evidence of grid-like representation patterns in the medial prefrontal cortex and entorhinal cortex. Moreover, the intensity of grid-like response scaled with the performance of navigating in social space and social avoidance trait scores. Our findings suggest a neurocognitive mechanism by which social information can be organized into a structured representation, namely cognitive map and its relevance to social well-being.

Some results on the Hamming distances of cyclic codes

Some results on the Hamming distances of cyclic codes

Maximum flag-rank distance codes

In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes.