Circulant Codes Research Articles

A method for constructing quaternary Hermitian self-dual codes and an application to quantum codes

We introduce quaternary modified four μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-circulant Hermitian self-dual codes. We also construct quaternary modified four μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.

Linear complementary dual codes and double circulant codes over a semi-local ring

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> be an odd prime power and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula> elements. In this paper, suppose ring <inline-formula><tex-math id="M4">\begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $\end{document}</tex-math></inline-formula> We first give a Gray map from <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_q^{4} $\end{document}</tex-math></inline-formula> and consider a decomposition of the ring <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Additionally, we investigate linear complementary dual (LCD) codes over the ring <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Some conditions for such linear codes over <inline-formula><tex-math id="M10">\begin{document}$ R $\end{document}</tex-math></inline-formula> to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula>.</p>

Double circulant codes for the Lee and Euclidean distance

<abstract><p>This paper investigates double circulant codes of length $ 2n $ over $ \mathbb{Z}_{{{p^m}}} $ where $ p $ is an odd prime, $ n $ goes to infinity, and $ m\ge 1 $ is a fixed integer. Using random coding, we obtain families of asymptotically good Lee codes over $ \mathbb{Z}_{{{p^m}}} $ in the case of small and large alphabets, and asymptotically good Euclidean codes over $ \mathbb{Z}_{{{p^m}}} $ for small alphabets. We use Euclidean codes to construct spherical codes, and Lee codes to construct insertion/deletion codes, by a projection technique due to (Yaglom, 1958) for spherical codes, and to (Sok et al., 2018) for deletion codes.</p></abstract>

Self-dual 2-quasi-cyclic codes and dihedral codes

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if charF=2), or a consta-dihedral code (if charF≠2). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. We also obtain necessary and sufficient conditions under which the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) of codes coincide with each other.

Thermal-Hydraulic System-Level Analysis of a Molten Salt Natural Circulation Loop

Molten salt reactors (MSRs) are a class of Generation IV nuclear reactors using molten salts as heat transfer fluids. MSRs bring a number of benefits, including low primary system working pressure, high working temperature, and enhanced safety due to the passive safety systems adopted. Although MSRs promise these benefits, a number of key technology needs, such as the accurate prediction of the thermal-hydraulic performance of the passive safety systems, which completely rely on natural circulation, are indispensable for MSR development, licensing, and future deployment. Therefore, this study develops the one-dimensional (1D) NAtural Circulation COde (NACCO) considering the buoyancy and radiative heat transfer effects in high-temperature molten salts for such predictions. The 1D code, developed using MATLAB, is then benchmarked with experimental data from three natural circulation flow experiments, where water, nitrate salt NaNO3-KNO3 (60–40 wt%), and fluoride salt LiF-BeF2 (66–34 mol%, FLiBe) were used as the working fluids. Our analysis shows that (1) the buoyancy and radiative heat transfer effects need to be considered for high-temperature molten salt natural circulation flows, while the radiative heat transfer effect is negligible for low-temperature water flows in the natural circulation experiments investigated, and (2) the 1D code NACCO predicts salt temperature profiles reasonably well, with less than 18°C and 25°C discrepancies from experimental data for the pipe centerline temperature of NaNO3-KNO3 and FLiBe up to 450°C and 750°C, respectively.

Self-dual and LCD double circulant codes over a class of non-local rings

Self-dual and LCD double circulant codes over a class of non-local rings

A DMST-based tool to establish the best aspect ratio, solidity and rotational speed for tidal turbines in real sea conditions

A Double Multiple Stream Tube (DMST) routine to predict the performance of cross-flow hydrokinetic turbines in real environments is presented, along with a site assessment application to identify the most efficient turbine aspect ratio, solidity and configuration (single, or paired) for a selected area of the Northern Adriatic Sea. The peculiarity of this DMST tool is its 3D character, since it allows to reproduce the vertical distribution of the torque generated by the turbine. To this end, correlations for fluid dynamic phenomena, based on high-fidelity fully CFD simulations, were implemented. The marine circulation code SHYFEM is adopted to obtain velocity profiles for a half lunar cycle period. The sites with the highest mean kinetic power were identified. The DMST routine is equipped with an iterative process able to establish which rotational speed maximizes the power output. Indeed, a spatially non-uniform velocity profile requires to determine the flow velocity more suitable to obtain the rotational speed via Tip Speed Ratio (TSR) definition. To this end, the section of the blades working at optimal TSR varies from top to bottom, until the maximum power is reached. It works as a virtual Maximum Power Point Tracking system able to adapt the turbine operating conditions for the different turbine geometries, and for changes in flow conditions. The results show that for the case study, the performance curve shape influences the optimal TSR blade section: the latter is often located in the upper part of the turbine for the low solidity, whereas, for high solidity turbines, in the bottom half part.

On the Computation of the Automorphisms Group of Some Optimal Codes Using Genetic Algorithm

Many research papers in coding theory have recently focused on designing high-rate codes or improving codes that exist through a better understanding and then improving the coding and decoding algorithm. As a result, this paper aims to investigate the computation of the Automorphisms groups of some optimal codes (e.g., some linear circulant codes where their distance meets the lower bound and nonlinear Nordstrom-Robinson (24, 28, 6) code). These Automorphisms groups provide information about the structure of the code, which aids in both the design and enhancement and improvement of decoding algorithms. A new genetic algorithm-based method is proposed, with a detailed description of its components, the fitness function, selection, crossover, and mutation, and is used to find an important collection of Automorphisms; the results obtained have shown that the proposed method is effective in finding stabilizers set for some types of codes.

An improved upper bound on self-dual codes over finite fields GF(11), GF(19), and GF(23)

This paper gives new methods of constructing symmetric self-dual codes over a finite field GF(q) where q is a power of an odd prime. These methods are motivated by the well-known Pless symmetry codes and quadratic double circulant codes. Using these methods, we construct an amount of symmetric self-dual codes over GF(11), GF(19), and GF(23) of every length less than 42. We also find 153 new self-dual codes up to equivalence: they are [32, 16, 12], [36, 18, 13], and [40, 20, 14] codes over GF(11), [36, 18, 14] and [40, 20, 15] codes over GF(19), and [32, 16, 12], [36, 18, 14], and [40, 20, 15] codes over GF(23). They all have new parameters with respect to self-dual codes. Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades.

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Let q be an odd prime power, and denote by $${\mathbb {F}}_q$$ the finite field with q elements. In this paper, we consider the ring $$R={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$ , where $$u^2=u, v^2=v,uv=vu=0$$ and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over $${\mathbb {F}}_5$$ as the Gray images of these codes over R are given in short lengths.

A Double Multiple Stream Tube (DMST) routine for site assessment to select efficient turbine aspect ratios and solidities in real marine environments

A MATLAB routine, based on a Double Multiple Stream Tube model, developed to quickly predict the performance of cross-flow hydrokinetic turbine, here is presented. The routine evaluate flow data obtained with the open-source marine circulation code SHYFEM. The tool can establish the best locations to place tidal devices taking into account bathymetric constraints and the hydrokinetic potential. Hence, it can be used to decide the best set of geometrical parameters. The geometrical variables of our analysis are turbine frontal area, aspect ratio and solidity. Several sub-models, validated with 3D and 2D CFD simulations, reproduce phenomena such as dynamic stall, fluid dynamic tips losses and the lateral deviation of streamlines approaching the turbine. As a case study, the tool is applied to an area of the northern Adriatic Sea. After having identified some suitable sites to exploit the energy resource, we have compared behaviours of different turbines. The set of geometrical parameters that gives the best performance in terms of power coefficient can vary considering several locations. Conversely, the power production is always greater for turbine with low aspect ratio (for a fixed solidity and area). Indeed, shorter devices benefit from higher hydrokinetic potentials at the top of the water column.

Embedding of a Blade-Element Analytical Model into the SHYFEM Marine Circulation Code to Predict the Performance of Cross-Flow Turbines

Our aim was to embed a 2D analytical model of a cross-flow tidal turbine inside the open-source SHYFEM marine circulation code. Other studies on the environmental impact of Tidal Energy Converters use marine circulation codes with simplified approaches: performance coefficients are fixed a priori regardless of the operating conditions and turbine geometrical parameters, and usually, the computational grid is so coarse that the device occupies one or few cells. In this work, a hybrid analytical computational fluid dynamic model based on Blade Element Momentum theory is implemented: since the turbine blades are not present in the grid, the flow is slowed down by means of bottom frictions applied to the seabed corresponding to forces equal and opposite to those that the blades would experience during their rotation. This simplified approach allowed reproducing the turbine behavior for both mechanical power generation and the turbine effect on the surrounding flow field. Moreover, the model was able to predict the interaction between the turbines belonging to a small cluster with hugely shorter calculation time compared to pure Computational Fluid Dynamics.

Construction of isodual codes from polycirculant matrices

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over $$ {\mathbb {F}}_2$$ in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over $${\mathbb {F}}_2,$$ we show that binary double polycirculant codes are asymptotically good.

Generalized Cyclotomic Double Circulant Self-Dual Codes of Length 4p

Constructions of some double circulant self-dual codes on the residue ring $Z_{2p}$ , where $p$ is a prime and satisfies $p \equiv 3 \bmod {4}$ , are proposed. Two new structures of these codes are formed from the theory of generalized cyclotomy. Moreover, both [12, 6, 4] and [44, 22, 8] optimal self-dual codes over GF(2) and a family of [28, 14, 9] highest known self-dual codes over GF(4) are given.

A modified Gilbert-Varshamov bound for self-dual quasi-twisted codes of index four

In this work, we study a family of self-dual four circulant codes and a family of self-dual four negacirculant codes, and the exact counting formula is derived for these families of codes. In addition, we prove that asymptotically good self-dual four circulant codes and negacirculant codes over finite fields exist, and both of them satisfy a modified Gilbert-Varshamov bound on the relative minimum distance with asymptotic rate 12.

On Self‐dual and LCD Double Circulant Codes over a Non‐chain Ring*

In this paper, we study double circulant codes of length 2n over the non-chain ring R = 𝔽 q + v𝔽 q + v 2𝔽 q ; where q is an odd prime power and v 3 = v. Exact enumerations of self-dual and LCD double circulant codes of length 2n over R are derived. When n is an odd prime, using random coding, we obtain families of asymptotically good self-dual and LCD codes of length 6n over 𝔽 q .

High resolution modeling of western Alaskan tides and storm surge under varying sea ice conditions

A depth-integrated high resolution (down to 25–50 m) tide and storm surge model has been developed for the Alaska region with a focus on the coasts of western Alaska. The model uses the ADCIRC basin-to-channel scale unstructured grid circulation code forced with Climate Forecast System version 2 reanalysis meteorology. The tidal solution has been validated at 121 shelf and nearshore stations, with lower overall errors than a data assimilated tidal model (FES2012). The ADCIRC model shows considerable skill at simulating tide and coastal surge, and resulting overland flooding on the Yukon–Kuskokwim Delta, during both summer and winter storms. The root-mean-square-error of the model does not exceed 20 cm at any station, outperforming the Global Ocean Forecasting System (GOFS 3.1), a 1/12∘ data assimilated ocean general circulation model which is coupled to sea ice physics. The ADCIRC model incorporates the effect of sea ice through parameterizations of the wind drag coefficient, modifying the air–sea momentum transfer under ice coverage. Three large winter storms with distinctly different ice coverages along the coasts of western Alaska were chosen to exhibit the variable effect of sea ice on the resulting storm surge. Under forming coastal ice coverage, local increases in water levels due to ice is seen in coastal areas, and under predominantly dense pack ice, an increase in momentum transfer in the marginal sea ice at the shelf break leads to an increase in sea levels across the entire Bering Sea, both commensurate with observations. Under the most variable ice fields, results are mixed, in part due to uncertainties in the air–sea–ice drag parameterization such as related to the assumption that sea ice drift and wind velocities are always well correlated. Including a description of sea ice drift and current speeds explicitly in the air–sea–ice drag formulation and considering wave–surge–ice interaction should improve the description of momentum transferred to the water column and resulting simulation of surge.

Assessment of the temporal evolution of storm surge across coastal Louisiana

The co-evolution of wetland loss and flood risk in the Mississippi River Delta is tested by contrasting the response of storm surge in coastal basins with varying historical riverine sediment inputs. A previously developed method to construct hydrodynamic storm surge models is employed to quantify historical changes in coastal storm surge. Simplified historical landscapes facilitate comparability while storm surge model meshes developed from historical data are incomparable due to the only recent (post-2000) extensive use of lidar for topographic mapping. Storm surge model meshes circa 1930, 1970 and 2010 are constructed via application of land to water (L:W) isopleths, lines that indicate areas of constant land to water ratio across coastal Louisiana. The ADvanced CIRCulation (ADCIRC) code, coupled with the Simulating WAves Nearshore (SWAN) wave model, is used to compute water surface elevations, time of inundation, depth-averaged currents and wave statistics from a suite of 14 hurricane wind and pressure fields for each mesh year. Maximum water surface elevation and inundation time differences correspond with coastal basins featuring historically negligible riverine sediment inputs and wetland loss as well as a coastal basin with historically substantial riverine inputs and wetland gain. The major finding of this analysis is maximum water surface elevations differences from 1970 to 2010 are 0.247 m and 0.282 m within sediment-starved Terrebonne and Barataria coastal basins, respectively. This difference is only 0.096 m across the adjacent sediment-abundant Atchafalaya-Vermilion coastal basin. Hurricane Rita inundation time results from 1970 to 2010 demonstrate an increase of approximately one day across Terrebonne and Barataria while little change occurs across Atchafalaya-Vermilion. The connection between storm surge characteristics and changes in riverine sediment inputs is also demonstrated via a sensitivity analysis which identifies changes in sediment inputs as the greatest contributor to changes in storm surge when compared with historical global mean sea level (GMSL) rise and the excavation of major navigation waterways. Results imply the magnitude of the challenge of preparing this area for future subsidence and GMSL rise.

On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$

Double circulant codes of length 2n over the non-local ring $R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,$ are studied when q is an odd prime power, and − 1 is a square in $\mathbb {F}_{q}$. Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over $\mathbb {F}_{q}$ are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal.

Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs

In this paper, we consider additive circulant graph codes which are self-dual additive \begin{document}$ \mathbb{F}_4 $\end{document} -codes. We classify all additive circulant graph codes of length \begin{document}$ n = 30, 31 $\end{document} and \begin{document}$ 34 \le n \le 40 $\end{document} having the largest minimum weight. We also classify bordered circulant graph codes of lengths up to 40 having the largest minimum weight.