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In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer n>3 divides the determinant (i 2 + d j 2) i 2 + d j 2 n 0 ≤ i , j ≤ (n − 1) / 2 , where d is any integer and (⋅ n) is the Jacobi symbol. Then we prove some divisibility results concerning |(i + dj)n|0≤i,j≤n−1 and |(i2 + dj2)n|0≤i,j≤n−1, where d ≠ 0 and n>2 are integers. Finally, for any odd prime p and integers c and d with p ∤ c d , we determine completely the Legendre symbol S c (d , p) p , where S c (d , p) := i 2 + d j 2 + c p 1 ≤ i , j ≤ (p − 1) / 2 . [ABSTRACT FROM AUTHOR]

Copyright of Journal of Chongqing University of Posts & Telecommunications (Natural Science Edition) is the property of Chongqing University of Posts & Telecommunications and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The secure electronic signature provides contracting parties, particularly the consumer, with safety and reassurance which has a favorable impact on business transactions due to the strong legal authority supplied by this signature, which is based on a method for its formation. Therefore, researchers are rushing to design safe and performance electronic signature schemes at the same time. We offer a novel signature technique based on two hard number theory issues in this work, Quadratic Residue (QR) and Chaotic Maps (CM). Several fields of study including mathematics, physics, and computer science have taken an interest in chaotic systems as a potential tool for cryptography. Analysis demonstrates that our strategy is more secure and efficient than other connected schemes, compared to other schemes. A proof of the proposed scheme's security against known key attacks is also provided in this article. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and let F p be the finite field of p elements. In 2019, Sun studied some permutations involving squares in F p . In this paper, by the theory of local fields we generalize this topic to F p 2 , which gives a partial answer to the question posed by Sun. [ABSTRACT FROM AUTHOR]

The authors analyze the special properties of two classes of quadratic and twisted Edwards curves over a prime field, which take into account their non-cyclic structure and the incompleteness of the point addition law. Both classes of curves contain singular points of 2nd and 4th orders with respect to one infinite coordinate, which generate points with uncertainty 0/0 in one of the coordinates of the sum, called fuzzy points. Five theorems are formulated and proved, which allow resolving these uncertainties and establishing the conditions whereby the point addition law in these classes of curves is complete. [ABSTRACT FROM AUTHOR]

A new fast public key cryptosystem is proposed, which is based on two dissimilar number-theoretic hard problems, namely the simultaneous chaotic maps (CM) problem and quadratic residue (QR) problem. The adversary has to solve the two hard problems simultaneously to recover the plaintext according to their knowledge about the public keys and the cipher-text. Cryptographic quadratic residue and chaotic system are employed to enhance the security of our cryptosystem scheme. The encryption, and decryption are discussed in details. Several security attacks are proposed to illustrate the system shield through chaotic maps and quadratic residue problems. The performance analysis of the proposed scheme show a much improved performance over existing techniques. [ABSTRACT FROM AUTHOR]

It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length 2 q + 2 contains the (0,1)-incidence matrix of a Hadamard 3- (2 q + 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). [ABSTRACT FROM AUTHOR]

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 ≡ 1 (mod 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. [ABSTRACT FROM AUTHOR]

In this letter, we study turbo product codes with quadratic residue codes (called QR-TPCs) as the component codes. We propose an efficient decoder based on Chase-II algorithm with two convergence conditions for the iterative decoding of QR-TPCs. For each row and column, the Chase-II decoder will stop immediately when one of the conditions is met. The simulation results show that the proposed algorithm has a lower computational complexity compared with existing decoding methods. Moreover, a comparison with 5G low-density parity-check codes shows that the proposed turbo product codes have better performance for short code lengths. [ABSTRACT FROM AUTHOR]

Traditional sound diffusers are quasi-random phase gratings attached to reflecting surfaces to introduce spatiotemporal incoherence into the backscattered acoustic field. Early designs consisted of periodically tiled diffuser grating unit cells to cover large surfaces. However, spatial periodicity leads to coherent constructive and destructive interference, which is undesirable for achieving acoustic diffusivity. This problem was partially addressed by using aperiodic tiling of unit cells based on pseudorandom sequences. While an aperiodic diffuser spacing can overcome the problems introduced by spatial periodicity, the improvements in performance come at the expense of increased thickness. In this work, we investigate spatiotemporal modulation of the surface acoustic admittance of a metasurface diffuser to improve sound diffusion. Using semi-analytical and finite element models, we demonstrate that the effects of the spatial periodicity can be mitigated without introducing an aperiodic spatial spacing, thus simultaneously minimizing diffuser thickness and improving diffusivity of the backscattered field. We develop a semi-analytical model that employs Fourier series expansion to determine the scattered sound field from a surface admittance consisting of a quadratic residue diffuser whose individual well admittances are modulated in a traveling wave fashion with modulation frequency, ω m , amplitude, Y m , and a wavenumber that matches the unit cell length, Λ. We observe significant improvement in diffusion due to the fact that the spatiotemporal modulation scatters sound into additional frequency-wavenumber pairs associated with harmonics of ω m and their diffraction orders. The semi-analytical model results are verified using a time-domain finite element model and compared with periodic and aperiodic diffuser designs. [ABSTRACT FROM AUTHOR]

Let S= \{ a_{1}, a_{2}, \dots, a_{n} \} be a finite set of non-zero integers. R. Balasubramanian, F. Luca and R. Thangadurai [Proc. Amer. Math. Soc. 138 (2010), pp. 2283–2288] gave an exact formula for the degree of the multi-quadratic field \mathbb {K}= \mathbb {Q}(\sqrt {a_{1}}, \sqrt {a_{2}}, \dots, \sqrt {a_{n}}) over \mathbb {Q}. In this paper, we calculate the explicit structure of the Galois group \operatorname {Gal}(\mathbb {K}/\mathbb {Q}) in terms of its action on \sqrt {a_{i}} for 1 \leq i \leq n. [ABSTRACT FROM AUTHOR]

Using pari gp, we analyze and give an implementation of the symmetric key cryptographic algorithm using the Paley graphs and binary codes. [ABSTRACT FROM AUTHOR]

A Z 2 s -additive code C of length n is a subgroup of Z 2 s n . Carlet (IEEE Trans Inf Theory 44:1543–1547, 1998) introduced a Gray map Φ (Carlet's Gray map) on Z 2 s and Tapia-Recillas and Vega (SIAM J Discret Math 17:103–113, 2003) have been shown that Φ (C) is a binary linear code if and only if for every u , v ∈ C , 2 (u ⊙ v) ∈ C , which their number is ∣ C ∣ 2 - ∣ C ∣ 2 . Let C be a linear code over Z 8 of type { δ 0 , δ 1 , δ 2 } . In this paper, by using the rows of the generator matrix (in standard form) of C , we introduce a set S with ∣ S ∣ ≤ δ 0 (δ 0 + 3) 2 + δ 1 and show that Φ (C) is linear if and only if for every u , v ∈ S , 2 (u ⊙ v) ∈ C . The results show that for a linear code C over Z 8 calculations to check the linearity of Φ (C) are less than Tapia-Recillas and Vega's method. In addition, for a cyclic code of odd length over Z 8 , a condition on the generator polynomial of this code is given which its Carlet's Gray image is linear. Also, as a result, we show that the Carlet's Gray image of a quadratic residue code over Z 8 is not linear. [ABSTRACT FROM AUTHOR]

Let p > 3 be a prime. Gauss first introduced the polynomial S p (x) = ∏ c (x − ζ p c) , where 0 < c < p and c varies over all quadratic residues modulo p and ζ p = e 2 π i / p . Later Dirichlet investigated this polynomial and used this to solve problems involving the Pell equations. Recently, Sun studied some trigonometric identities involving this polynomial. In this paper, we generalize their results. As applications of our result, we extend Chowla's result on the congruence concerning the fundamental unit of ℚ (p) and give an equivalent form of the extended Ankeny et al. conjecture. [ABSTRACT FROM AUTHOR]

In this paper, a low-complexity decoder based on a neural network is proposed to decode binary quadratic residue (QR) codes. The proposed decoder is based on the neural min-sum algorithm and the modified random redundant decoder (mRRD) algorithm. This new method has the same asymptotic time complexity as the min-sum algorithm, which is much lower than the difference on syndromes (DS) algorithm. Simulation results show that the proposed algorithm achieves a gain of more than 0.4 dB when compared to the DS algorithm. Furthermore, a simplified approach based on trapping sets is applied to reduce the complexity of the mRRD. This simplification leads to a slight loss in error performance and a reduction in implementation complexity. [ABSTRACT FROM AUTHOR]

The security and robustness of the existing cryptographic models are ensured, thanks to algebraic or arithmetic problems such as the factorization problem, the discrete logarithm, and the quadratic residues. In this article, we consider the problem of performing sequence of local complementations on the Paley graphs. We then propose an effective novel public-key cryptosystem based on this problem. [ABSTRACT FROM AUTHOR]

Fix a prime p>2$p >2$ and a finite field Fq$\mathbb {F}_{q}$ with q elements, where q is a power of p. Let m be a monic polynomial in the polynomial ring Fq[T]$\mathbb {F}_{q}[T]$ such that deg(m)$\deg (m)$ is large. Fix an integer r⩾2$r\geqslant 2$, and let a1,⋯,ar$a_1,\dots ,a_r$ be distinct residue classes modulo m that are relatively prime to m. In this paper, we derive an asymptotic formula for the natural density δm;a1,⋯,ar$\delta _{m;a_1,\dots ,a_r}$ of the set of all positive integers X such that ∑N=1Xπq(a1,m,N)>∑N=1Xπq(a2,m,N)>⋯>∑N=1Xπq(ar,m,N)$\sum _{N=1}^{X} \pi _{q}(a_1,m,N) > \sum _{N=1}^{X} \pi _{q}(a_2,m,N)>\cdots > \sum _{N=1}^{X} \pi _{q}(a_r,m,N)$, where πq(ai,m,N)$\pi _{q}(a_i,m,N)$ denotes the number of irreducible monic polynomials in Fq[T]$\mathbb {F}_{q}[T]$ of degree N that are congruent to aimodm$ a_i \bmod m$, under the assumption of LI (Linear Independence Hypothesis). Many consequences follow from our results. First, we deduce the exact rate at which δm;a1,a2$\delta _{m;a_1,a_2}$ converges to 12$\frac{1}{2}$ as deg(m)$\deg (m)$ grows, where a1 is a quadratic non‐residue and a2 is a quadratic residue modulo m, generalizing the work of Fiorilli and Martin. Furthermore, similarly to the number field setting, we show that two‐way races behave differently than races involving three or more competitors, once deg(m)$\deg (m)$ is large. In particular, biases do appear in races involving three or more quadratic residues (or quadratic non‐residues) modulo m. This work is a function field analog of the work of Lamzouri, who established similar results in the number field case. However, we exhibit some examples of races in function fields where LI is false, and where the associated densities vanish, or behave differently than in the number field setting. [ABSTRACT FROM AUTHOR]

Let p be a prime and F p be the set of integers modulo p. Let χ p be a function defined on F p such that χ p (0) = 0 and for a ∈ F p \ { 0 } , set χ p (a) = 1 if a is a quadratic residue modulo p and χ p (a) = − 1 if a is a quadratic non-residue modulo p. Note that χ p (a) = a p is indeed the Legendre symbol. The image of χ p in the set of real numbers. In this paper, we consider the following sum ∑ x ∈ F p χ p ((x − a 1) (x − a 2) ... (x − a t)) where a 1 , a 2 , ... , a t are distinct elements in F p . [ABSTRACT FROM AUTHOR]

This paper deals with the imaging problem for sliding spotlight (SS) and terrain observation by progressive scan (TOPS) modes in bistatic configuration of the geostationary (GEO) transmitter with a low earth orbit satellite (LEO) receiver, named GTLR-BiSAR system. A unified imaging algorithm is proposed to process the GTLR-BiSAR data acquired in SS or TOPS modes. Our main contributions include four aspects. Firstly, the imaging geometry of this novel configuration is described in detail. Furthermore, the GTLR-BiSAR signal expressions were deduced in both time and frequency domains. These signal expressions provide great support for the design of processing the algorithm theoretically. Secondly, we present a unified deramping-based technique according to the special geometry of GTLR-BiSAR to overcome the azimuth spectrum aliasing phenomenon, which typically affects SS and TOPS data. Thirdly, the spatial variance of GTLR-BiSAR data were thoroughly analyzed based on the range-Doppler (RD) geolocation functions. On the basis of a former analysis, we put forward the azimuth variance correction strategy and modified the conventional chirp scaling function to solve the range variance problem. Finally, we completed the derivation of the two-dimensional spectrum after the range chirp scaling. On the basis of spectrum expressions, we compensated for the quadratic and residue phase, and the azimuth compression was completed by SPECAN operation. In addition, we provide a flow diagram to visually exhibit the processing procedures. At the end of this paper, the simulation and real data experiment results are presented to validate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce double Quadratic Residue Codes (QRC) of length n = p + q for prime numbers p and q in the ambient space F 2 p × F 2 q. We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when p = q. We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting. [ABSTRACT FROM AUTHOR]

Grid impedance has a significant impact on the small-signal stability and control of grid-connected power converters used for connecting multiple distributed energy resources and in various motor drive applications. To apply advanced control schemes, such as adaptive control, real-time grid impedance is a necessary parameter that ensures stability margins of the system during operation. For real-time grid-impedance measurement purpose, broadband frequency-response measurement techniques based on the pseudorandom binary sequence can be used. In this article, a computationally effective perturbation signal quadratic residue binary (QRB) sequence is proposed for real-time grid impedance measurement. Unlike the conventional perturbation signals, the QRB sequence has low crest factor and less perturbation duration resulting in minimal disturbance during impedance measurement. To extract the grid impedance in real-time, the QRB sequence is added to the reference d-axis and q-axis currents of the battery energy storage system (BESS). Next, perturbed voltage and current signals are extracted in the abc-frame and converted to the d-q-frame, and grid impedance is assessed. Subsequently, the system stability is analyzed using the generalized Nyquist stability criterion. Later, an adaptive control algorithm is proposed and applied to the energy storage system (ESS) to enhance stability margins of the system based on the measured grid impedance. The effectiveness of the proposed perturbation sequence for impedance measurement and stability improvement algorithm are tested on a standard CIGRE microgrid system in the MATLAB/Simulink simulation environment. Experiments are conducted on a reduced scale 3 kVA laboratory testbed and results are presented to validate the efficacy of the proposed perturbation sequence and stability improvement algorithm. [ABSTRACT FROM AUTHOR]

For a prime p ≥ 5 p ≥ 5 let q 0 , q 1 , ... , q (p − 3) / 2 q 0 , q 1 , ... , q (p − 3) / 2 be the quadratic residues modulo p p in increasing order. We study two (p − 3) / 2 (p − 3) / 2 -periodic binary sequences (d n) (d n) and (t n) (t n) defined by d n = q n + q n + 1 mod 2 d n = q n + q n + 1 mod 2 and t n = 1 t n = 1 if q n + 1 = q n + 1 q n + 1 = q n + 1 and t n = 0 t n = 0 otherwise, n = 0 , 1 , ... , (p − 5) / 2 n = 0 , 1 , ... , (p − 5) / 2. For both sequences we find some sufficient conditions for attaining the maximal linear complexity (p − 3) / 2 (p − 3) / 2. Studying the linear complexity of (d n) (d n) was motivated by heuristics of Caragiu et al. However, (d n) (d n) is not balanced and we show that a period of (d n) (d n) contains about 1 / 3 1 / 3 zeros and 2 / 3 2 / 3 ones if p p is sufficiently large. In contrast, (t n) (t n) is not only essentially balanced but also all longer patterns of length s s appear essentially equally often in the vector sequence (t n , t n + 1 , ... , t n + s − 1) (t n , t n + 1 , ... , t n + s − 1) , n = 0 , 1 , ... , (p − 5) / 2 n = 0 , 1 , ... , (p − 5) / 2 , for any fixed s s and sufficiently large p p. [ABSTRACT FROM AUTHOR]

Let p > 3 be a prime, and let ( · p ) be the Legendre symbol. Let b ∈ Z and ε ∈ {±1}. We mainly prove that l{Np (a,b) : 1 < a < p and (a/p) = ε }l = ( 3−( −1 p ) )/ 2, where Np (a,b) is the number of positive integers x < p/2 with {x 2 +b}p > {ax 2 +b}p, and {m}p with m ∈ Z is the least nonnegative residue of m modulo p. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate various kinds of congruences related to the Wilson quotient by applying a Miki-type linear identity involving two different kinds of sums for Bernoulli numbers, and further by reecting characteristics of quadratic residues and non-residues modulo p, where p is a prime number satisfying p ≡ 1 (mod 4). [ABSTRACT FROM AUTHOR]

Copyright of Revista Integración is the property of Universidad Industrial de Santander and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs. [ABSTRACT FROM AUTHOR]

Using elementary methods, we count the quadratic residues of a prime number of the form p = 4 n − 1 in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number h of the imaginary quadratic field Q (− p). Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found. [ABSTRACT FROM AUTHOR]

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

Copyright of Journal of Chongqing University of Posts & Telecommunications (Natural Science Edition) is the property of Chongqing University of Posts & Telecommunications and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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. [ABSTRACT FROM AUTHOR]

In this paper, we present the Neutrosophic quadratic residues and nonresidues with their basic interpretation as graphs in an algebraic manner and analog to the algebraic graphs. We establish the Neutrosophic, number-theoretic, and graph-theoretic properties of the set of Neutrosophic quadratic residues and nonresidues, many of which mirror those of the classical quadratic residues and nonresidues of modulo an odd prime. These properties, especially the algebraic ones, are connected to algebraic graphs, and thus we conclude the paper by studying the structural properties of Neutrosophic quadratic residue and quadratic nonresidue graphs. [ABSTRACT FROM AUTHOR]

This study examined (in)congruences between beliefs and practices of EFL university teachers on in‐class oral corrective feedback (OCF). The participants were 20 university English language teachers from a private university in Turkey. Data were collected via video‐recorded non‐participant detached observation, a task about OCF to determine the beliefs of the teachers, and a stimulated recall interview. The results showed incongruence between what the teachers said they believed and what they did. However, teachers' beliefs and practices were similar regarding whether the errors should be corrected, when errors should be corrected, and who should correct them. Particularly notable in this study was the finding that those teachers with the greatest incongruence almost always stood by their decisions, even after they watched their unsuccessful OCF practices. [ABSTRACT FROM AUTHOR]

In this paper, we study quadratic residue (QR) codes of prime length q over the ring R = p + u p + v p with u 2 = u , v 2 = v and u v = v u = 0 , where p and q are distinct odd prime numbers. We analyze some basic properties of cyclic codes of length n over R , we define QR codes by their generating idempotents. Further, we discuss the extended QR codes. We present a considerable number of good p -ary codes as Gray images of QR codes over p + u p + v p by considering the case when p is an odd prime. [ABSTRACT FROM AUTHOR]

We prove that if ε(m) → 0 arbitrarily slowly, then for almost all m and any A ⊂ Zm such that A−A does not contain non-zero quadratic residues we have |A| ≤ m1/2−ε(m). [ABSTRACT FROM AUTHOR]

In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length nr from cyclic codes of length n and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this way. Second, in the other direction, we show that for cyclic codes of length n with a primitive n-th root of unity as a non-zero, the minimum distance is roughly bounded by the square-free part of n. This means that we shall not expect good cyclic codes of length n in general if, for example, the length n is divisible by a high power of a prime. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A step forward is made in a long standing Lovász problem regarding existence of Hamilton paths in vertex-transitive graphs. It is shown that a vertex-transitive graph of order a product of two primes arising from a primitive action of PSL(2;p) on the cosets of a subgroup isomorphic to Dp−1 has a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton cycles, such as Chvátal's theorem and Jackson's theorem, to certain results from matrix algebra, graph quotienting, and polynomial representations of quadratic residues in terms of primitive roots in finite fields. Also, Hamilton cycles are proved to exist in vertex-transitive graphs of order a product of two primes arising from a primitive action of either PΩ(2d; 2), M22, A7, PSL(2; 13), or PSL(2; 61). The results of this paper, combined together with other known results, imply that all connected vertex-transitive graphs of order a product of two primes, except for the Petersen graph, have a Hamilton cycle. [ABSTRACT FROM AUTHOR]

• The insertion loss of noise barrier with MPP-QRD top structure are measured on site. • The impedance of MPP and QRD structure was calculated respectively to analyze the matching degree. • The acoustic impedance matching degree is significantly correlated with the improvement in noise barrier performance. • The MPP-QRD top structure with acoustic impedance matching can significantly enhance the insertion loss of noise barrier. The effective edge processing is the key to the performance of noise barrier. To prevent the sound diffraction within the low-medium frequency range, the quadratic residue diffuser (QRD) was combined with microperforated panel (MPP) in this study. The impedance of MPP with different parameters and QRD structure with N = 7 was calculated respectively and the matching degree of the composite structure was obtained. The insertion loss of noise barriers with different MPP-QRD top structures was field measured and compared to the equi-height vertical noise barrier. The results show that the degree of impedance matching between microperforated panels and quadratic residue diffusers is significantly correlated with their performance as the top structure of noise barrier. The acoustic impedance matched MPP-QRD top structure can significantly improve the insertion loss of the vertical noise barrierwith the same height. When the microperforated panel perforation is 3 %, the thickness is 0.8 mm, and the aperture is 0.6 mm, performance of the MPP-QRD structure is the best, the relative insertion loss can reach 4.07 dB. [ABSTRACT FROM AUTHOR]

The codewords of weight 10 of the [42, 21, 10] extended binary quadratic residue code are shown to hold a design of parameters 3−(42,10,18). Its automorphism group is isomorphic to PSL(2,41). Its existence can be explained neither by a transitivity argument, nor by the Assmus–Mattson theorem. [ABSTRACT FROM AUTHOR]

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that α(...(1, 2)) = 8q+5, and for the diachromatic number we state that dac(...( 1, 2)) = 8q + 3. In general, we give the lower bounds α(... (1, a)) 1; a)) ≥ 4q + 1 and dac(... (1, a)) ≥ 4q+3 when a is a non quadratic residue of ℤ4q+1 for graphs and ℤ4q+3 for digraphs, and the equality is attained, in both cases, for a = 3. Finally, we determine the achromatic index for circulant graphs of q² + q + 1 vertices when the projective cyclic plane of odd order q exists. [ABSTRACT FROM AUTHOR]

Multilevel and compartmented access structures are two important classes of access structures where participants are grouped into levels/compartments with different degrees of trust and privileges. The construction of secret sharing schemes for such access structures has been the attention of researchers for a long time. Two main approaches have been taken so far, one of them is based on polynomial interpolation and the other one is based on the Chinese Remainder Theorem (CRT). In this article the first asymptotically ideal CRT‐based secret sharing schemes for (disjunctive, conjunctive) multilevel and compartmented access structures are proposed. Our approach is compositional and it is based on a variant of the Asmuth‐Bloom secret sharing scheme where some participants may have public shares. Based on this, the proposed secret sharing schemes for multilevel and compartmented access structures are asymptotically ideal if and only if they are based on 1‐compact sequences of co‐primes. Possible applications for secret image and multi‐secret sharing are pointed‐out. [ABSTRACT FROM AUTHOR]

The quadratic residuosity problem is the problem to distinguish between the dis-tributions of quadratic residues and quadratic non-residues modulo a composite integer. The intractability of the quadratic residuosity problem is the basis for the security of numerous constructions in cryptography, including public-key en-cryption schemes, pseudo-random generators, or cryptographic protocols. The aim of this talk is to provide a brief overview on the quadratic residuosity problem and its applications to cryptography. [ABSTRACT FROM AUTHOR]

If this were not the case, then we would have two points I P SB i sb i and I P SB j sb i that have the same distance from line Graph HT ht and are on the same side of that line. Now consider an element Graph HT ht . Determine all points I X i for which the area of triangle Graph HT ht is minimized. A variable point I X i is chosen on minor arc Graph HT ht of I i , and segments Graph HT ht and Graph HT ht meet at I D i . [Extracted from the article]

The proposal of the ubiquitous power Internet of Things (UPIoT) has increased the demand for communication coverage and data collection of smart grid; the quantity and quality of communication networks are facing greater challenges. This brief applies (73, 37, 13) quadratic residue (QR) codes to power line carrier technology to improve the quality of local data communication in UPIoT. In order to improve the decoding performance of the QR codes, an induction method for the error pattern is proposed, which can divide the originally coupled error pattern into six parts and reuse the same module for decoding. This method greatly reduces the resource requirements, so that (73, 37, 13) QR code can be implemented on FPGA hardware. Notably, the hardware architecture is a modular framework, which can fit into an FPGA with different sizes. As an example (73, 37, 13), QR code is implemented on Intel Arria10 FPGA; the experimental result shows that the maximum decoding frequency of this architecture is 21.7 M Hz, which achieves 4121x speedup compared to CPU. Moreover, the proposed architecture benefits from high flexibility, such as modular design and decoding framework in the form of the pipeline which can be seen as an alternative scheme for decoding long-length QR codes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove that the clique number of the Paley graph is at most p/2+1, and that any supposed additive decompositions of the set of quadratic residues can only come from co‐Sidon sets. [ABSTRACT FROM AUTHOR]