Quadratic Residue Research Articles

On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length $$2q+2$$ contains the (0,1)-incidence matrix of a Hadamard 3- $$(2q+2,q+1,(q-1)/2)$$ design D(q) associated with a Paley–Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley–Hadamard matrix of type I. If $$q=5, 11, 17, 23$$ , then the full permutation automorphism group of L(q) coincides with the full automorphism group of D(q), and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code C(17) are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley–Hadamard matrix H of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix $$H'$$ such that the symmetric 2-(36, 15, 6) design D associated with $$H'$$ has trivial full automorphism group, and the incidence matrix of D spans a ternary code equivalent to C(17).

Comparison Failure and Successful Methodologies for Diffusion Measurements Undertaken inside Two Different Testing Rooms

The scattering phenomenon is known to be of great importance for the acoustic quality of a performance arts space. The scattering of sound can be achieved in different ways: it can be obtained by the presence of architectural and/or decorating elements inside a room (e.g., columns, statues), by the geometry and roughness of a surface (e.g., Quadratic Residue Diffuser (QRD)) and by the diffraction effect occurring when a sound wave hits the edges of an obstacle. This article deals with the surface scattering effects and the diffusion phenomenon only related to MDF and plywood panels tested by disposing the wells both horizontally and vertically. The test results undertaken inside a semi-reverberant room and inside a large reverberant room have been compared to highlight the success and the failure of the measuring methodologies. In detail, according to the existing standards and regulations (i.e., ISO 17497—Part 2), diffusion measurements have been undertaken on a few selected types of panel: two QRD panels (made of Medium Density Fiberboard (MDF) and plywood) with and without a smooth painted solid wood placed behind the QRD. The panels have been tested inside two rooms of different characteristics: a semi-anechoic chamber (Room A) and a large reverberant room (Room B). The volume size influenced the results that have been analyzed for both chambers, showing an overlap of reflections on panels tested inside Room A and a clear diffusion response for the panels tested inside Room B. In terms of the diffusion coefficient in all the octave bands between 125 Hz and 8 kHz, results should not be considered valid for panels tested in Room A because they were negatively impacted by extraneous reflections, while they are reliable for panels tested in Room B.

Stabilization of the extended finite element method for stiff embedded interfaces and inclusions

AbstractThis article investigates the performance of the XFEM for stiff embedded interfaces and inclusions. It is known that the XFEM may lead to ill‐conditioned stiffness matrices and oscillations in the interface traction field, the severity of which depends on the underlying basis functions, as well as on the orientation of the interface. The jumps at the discontinuity are shown to contain quadratic bubble residuals that can introduce oscillatory behavior. Those residuals are worsened by the ill‐conditioning of the system with very stiff interfaces. A variationally consistent method is proposed to overcome the oscillatory behavior and ill‐conditioning, in which the assumed strain method is developed to directly eliminate bubble residuals and deploy Legendre polynomials and explore their orthogonality properties to improve the conditioning of the stiffness matrices. Numerical examples illustrate the efficiency and generality of the proposed approach at both element and structural levels. The new approach is shown to be robust for imposing Dirichlet type boundary conditions at the crack interface, such as crack closure and initially rigid cohesive laws. The effects of numerical oscillations on the prediction of effective properties of composites in XFEM‐based computational homogenization procedures are also discussed.

Machine learning for decoding linear block codes: case of multi-class logistic regression model

<p>Facing the challenge of enormous data sets variety, several machine learning-based algorithms for prediction (e.g, Support vector machine, multi layer perceptron and logistic regression) have been highly proposed and used over the last years in many fields. Error correcting codes (ECCs) are extensively used in practice to protect data against damaged data storage systems and against random errors due to noise effects. In this paper, we will use machine learning methods, especially multi-class logistic regression combined with the famous syndrome decoding algorithm. The main idea behind our decoding method which we call logistic regression decoder (LRDec) is to use the efficient multi-class logistic regression models to find errors from syndromes in linear codes such as bose, ray-chaudhuri and hocquenghem (BCH), and the quadratic residue (QR). Obtained results of the proposed decoder have a significant benefit in terms of bit error rate (BER) for random binary codes. The comparison of our decoder with many competitors proves its power. The proposed decoder has reached a success percentage of 100% for correctable errors in the studied codes.</p>

Improved sound diffuser performance using spatiotemporally modulated metasurfaces

Conventional sound diffuser designs consist of a periodic arrangement of variable depth wells which result in a quasi-random spatial distribution of the reflected phase and minimal energy loss. Diffuser performance depends on geometric parameters such as well depth, number of elements per diffuser period, and in-plane dimensions. We previously showed that coiled-space acoustic metasurfaces (AMS) can significantly reduce diffuser depth while maintaining the performance when compared to classical quadratic residue diffusers (QRD) [Kang and Haberman, J. Acoust. Soc. Am., 149(4), A141 (2021)]. However, all conventional and AMS diffuser performance is limited by geometry due to the linear relationship between frequency and wavelength. Here we consider spatiotemporal modulation of an AMS to control and improve sound diffusion using a technique inspired by nonreciprocal vibrations in Euler beams [Goldsberry et al., Phys. Rev. B 102 (2020)]. A theoretical model is developed using the classical grating equation augmented with a surface impedance that is modulated as a travelling wave function. Properly selected modulation parameters lead to increased diffusion coefficients associated with scattering into harmonics of the temporal modulation frequency and nonreciprocal scattering patterns dependent on the spatiotemporal modulation. Diffuser performance is demonstrated and investigated through the angular dependence of the scattered field magnitude.

An experimental investigation into the scattering properties of maximum length sequence diffusers and quadratic residue diffusers

Since the invention of Schroeder diffusers, extensive studies have been carried out on the prediction and measurement of diffusion coefficients and correlation scattering coefficients in the far field, providing valuable referring data for public. However, as another important evaluation index, the random-incidence scattering coefficients of Schroeder diffusers, which are popularly used in geometric room acoustic models for accurate predictions, have not yet been systematically studied in the diffusing field. This paper presents an experimental study on the scattering coefficients of two well-known Schroeder diffusers (maximum length sequence (MLS) diffusers and quadratic residue diffusers (QRDs)) in one 1:10 scaled-down reverberation room. The scattering coefficients of these Schroeder diffusers were compared to discover their variation laws as a function of well depth, well width, sequence length and the ratio of the maximum well depth to the wave length. To validate the accuracy and reliability of the scaled-down measurements, the scattering coefficients of six MLS diffusers, as well as their standard deviations, were also measured in a full-size reverberation room. The results measured in this study provide engineers and acousticians with some useful reference data in room acoustics design.

Quadratic residue codes, rank three groups and PBIBDs

The automorphism group of the Zetterberg code Z of length 17 (also a quadratic residue code) is a rank three group whose orbits on the coordinate pairs determine two strongly regular graphs equivalent to the Paley graph attached to the prime 17. As a consequence, codewords of a given weight of Z are the characteristic vectors of the blocks of a PBIBD with two associate classes of cyclic type. More generally, this construction of PBIBDs is extended to quadratic residue codes of length \(\equiv 1 \pmod {8},\) to the adjacency codes of triangular and lattice graphs, and to the adjacency codes of various rank three graphs. A remarkable fact is the existence of 2-designs held by the quadratic residue code of length 41 for code weights 9 and 10.

Design and verification of ultrathin Schroeder diffuser based on membrane-type resonators

We present the design and experimental verification of a membrane-type resonator-based Schroeder diffuser (MRSD). The membrane-type resonator consists of a decorated membrane and a rigid back cavity. By combining multiple resonators with the phase shift profile based on the quadratic residue sequence, the MRSD can produce a high efficiency diffuse sound field. Furthermore, the designed MRSD has a thickness with deep-subwavelength scale, which is 1 order of magnitude thinner than the conventional diffuser constructed with periodical wells. The proposed MRSD has advantages in low frequency applications and offers versatility for novel manipulations of sound.

New extremal binary self-dual codes from block circulant matrices and block quadratic residue circulant matrices

In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2 and F2+uF2. Using extensions, neighbours and sequences of neighbours, we construct many new self-dual codes. In particular, we construct one new self-dual code of length 66 and 51 new self-dual codes of length 68.

A residual-based test for multivariate GARCH models using transformed quadratic residuals

This paper provides a residual-based approach to examine the adequacy of multivariate GARCH models. We employ the transformed quadratic residuals to construct the residual-based statistic and derive its correct asymptotic distribution by taking into account the impact of parameter estimation uncertainty. The simulation results indicate that the residual-based test achieves reasonable sizes and comparable powers. An empirical application further shows the usefulness of the proposed test.

Sets whose differences avoid squares modulo 𝑚

We prove that if ε ( m ) → 0 \varepsilon (m)\to 0 arbitrarily slowly, then for almost all m m and any A ⊂ Z m A\subset \mathbb {Z}_m such that A − A A-A does not contain non-zero quadratic residues we have | A | ⩽ m 1 / 2 − ε ( m ) . |A|\leqslant m^{1/2-\varepsilon (m)}.

A conjecture of Sárközy on quadratic residues

For any prime p, let Rp be the set of all quadratic residues modulo p. In 2012, Sárközy proved that if p is a sufficiently large prime, then Rp has no 3-decomposition A+B+C=Rp with |A|,|B|,|C|≥2. In this paper, we prove that for any prime p, Rp has no additive 3-decomposition A+B+C=Rp with |A|,|B|,|C|≥2. Furthermore, for any prime p, if A+B=Rp is a 2-decomposition, then 0.17p+1<|A|,|B|<2.8p−6.63. We also pose three related conjectures.

Generalized Galbraith’s Test: Characterization and Applications to Anonymous IBE Schemes

The main approaches currently used to construct identity-based encryption (IBE) schemes are based on bilinear mappings, quadratic residues and lattices. Among them, the most attractive approach is the one based on quadratic residues, due to the fact that the underlying security assumption is a well-understood hard problem. The first such IBE scheme was constructed by Cocks, and some of its deficiencies were addressed in subsequent works. In this paper, we focus on two constructions that address the anonymity problem inherent in Cocks’ scheme, and we tackle some of their incomplete theoretical claims. More precisely, we rigorously study Clear et al.’s and Zhao et al.’s schemes and give accurate probabilities of successful decryption and identity detection in the non-anonymized version of the schemes. Furthermore, in the case of Zhao et al.’s scheme, we give a proper description of the underlying security assumptions.

Counting certain quadratic partitions of zero modulo a prime number

Abstract Consider an odd prime number p ≡ 2 ( mod 3 ) p\equiv 2\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) . In this paper, the number of certain type of partitions of zero in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} with all three parts chosen from the set of non-zero quadratic residues mod p p . Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation x 2 + y 2 + z 2 = 0 {x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} . Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.

The extended binary quadratic residue code of length 42 holds a 3‐design

AbstractThe codewords of weight 10 of the [42, 21, 10] extended binary quadratic residue code are shown to hold a design of parameters . Its automorphism group is isomorphic to . Its existence can be explained neither by a transitivity argument, nor by the Assmus–Mattson theorem.

Experimental demonstration of a two-dimensional sound diffusing acoustic metasurface with subwavelength thickness

Acoustic diffusers are surfaces that minimize the strength of room modes in small- and medium-sized rooms and reduce specular reflections in large spaces. Since diffusers are surface treatments that are mounted to walls of interior spaces, it is desirable to reduce their thickness in order to maximize usable space. Traditional diffuser designs, such as the quadratic residue diffuser (QRD) make use of variable depth wells to control the local phase of the reflected field. In these designs, well depth is directly proportional to the reflected phase, rendering the overall thickness dependent on the design frequency. Coiled-space (CS) structures are good candidates to minimize diffuser thickness while maintaining performance. However, CS structures have tortuous channels which are susceptible to viscous losses which can degrade diffuser performance. This work presents a CS design for a 2D QRD (N = 7) diffuser with minimal viscous losses and λ/8 thickness, a 4x reduction compared to traditional QRD. Six unit-cell designs were optimized, fabricated, and measured in an impedance tube. Measured reflection coefficients are in a good agreement with predictions. The 3D diffusion performance of the CS and traditional diffusers were measured on multiple planes orthogonal to the diffuser surface with CS and traditional designs performing comparably.

Skyline quadratic residue diffuser-resonator (SQRD-R): a study about its characteristic

To create a comfortable acoustic environment both indoor and outdoor, we need to control the noise. In the room, noise comes from both outside and inside the room for example is flatter echo. Installation of acoustic material such as absorber, resonator, or diffuser is an example of indoor noise control. QRD is one of diffuser consider of wells with different depth to diffuse the reflected sound. The depth of the wells is calculated by DR. Manfred Schroeder formula. In this research, a combination of Skyline QRD and Helmholtz resonator is given, we call it SQRD-R. This SQRDR follows the design from Wes Lachot Design but using a Helmholtz resonator as the cell. The examination of its characteristic is built using both of impedance tube test and calculation. The purposes of this research are learning about the effect of holes position on its sound impedance and sound absorption coefficient. The highest impedance value is –j(3.5 x 107) acoustic ohm. The highest absorption coefficient is 0.9. By controlling the SQRD-R dimension, we can control noise at specific frequency.

On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums

The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime p to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.

Edge even and edge odd graceful labelings of Paley Graphs

Edge even graceful labeling is a novel graceful labelling, introduced in 2017 by Elsonbaty and Daoud. A graph G with p vertices and q edges is called an edge even graceful if there is a bijection f: E(G) → {2, 4,., 2q} such that, when each vertex is assigned the sum of the labels of all edges incident to it mod 2k, where k = max (p, q), the resulting vertex labels are distinct. A labeling of G is called edge odd graceful labeling, if there exists a bijection f from the set of edges E(G) to the set {1,3,5,…,2q-1} such that the induced the map f* from the set of vertices V(G) to {0,1,2,.,.,2q-1} given by f*(u) = Σ uv∈E(G) f(uv) (mod 2q) is an injection. A graph which admits edge even (odd) graceful labeling is called an edge even (odd) graceful graph. Paley graphs are dense undirected graphs raised from the vertices as elements of an appropriate finite field by joining pairs of vertices that differ by a quadratic residue. In this paper, we study the construction of edge even (odd) graceful labeling for Paley graphs and prove that Paley graphs of prime order are edge even (odd) graceful.

Symmetries of abelian Chern-Simons theories and arithmetic

We determine the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern-Simons theories. The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern-Simons levels, revealing interesting connections with number theory. We give a complete characterization of the symmetries of abelian topological field theories and along the way find many theories that are non-trivially time-reversal invariant by virtue of a quantum symmetry, including U(1)k Chern-Simons theory and (ℤk)ℓ gauge theories. For example, we prove that U(1)k Chern-Simons theory is time-reversal invariant if and only if −1 is a quadratic residue modulo k, which happens if and only if all the prime factors of k are Pythagorean (i.e., of the form 4n + 1), or Pythagorean with a single additional factor of 2. Many distinct non-abelian finite symmetry groups are found.