Linear codes with balanced weight distribution

In this paper, we study particular linear codes defined overF _q , with an astonishing property, their weight distribution is balanced, i.e. there is the same number of codewords for each nonzero weight of the code. We call these codesBWD-codes. We first study BWD-codes by means of the Pless identities and we completely characterize the two-weight projective case. We study the class of codes defined under subgroups of the multiplicative group ofF _q ^s , using the Gauss sums. Then, given a primep and an integerN dividingp – 1, we construct all theN-weight BWD-codes of that class. We conclude this paper by some tables of BWD-codes and an open problem.     

On cyclic codes which areq-ary images of linear codes

This paper presents a complete characterization of cyclic codes over GF(q) which areq-ary images of linear codes over GF(q ²). New cyclic codes over GF(2 ^r ) are constructed as images of other cyclic codes over GF(2^3r ), for some positive integersr. An application to decoding is given.     

Computation of the weight distribution of CRC codes

In this article, we illustrate an algorithm for the computation of the weight distribution of CRC codes. The recursive structure of CRC codes will give us an iterative way to compute the weight distribution of their dual codes starting from some “representative” words. Thanks to MacWilliams’ Theorem, the computation of the weight distribution of the dual codes can be easily brought back to that of CRC codes. The author was partially supported by the Swiss National Science Foundation under Grant no. 113251.     

Bounds on the minimum distance of the duals of extended BCH codes over\mathbb{F}_p

LetC be an extended cyclic code of lengthp ^m over . The border ofC is the set of minimal elements (according to a partial order on [0,p ^m –1]) of the complement of the defining-set ofC. We show that an affine-invariant code whose border consists of only one cyclotomic coset is the dual of an extended BCH code if, and only if, this border is the cyclotomic coset, sayF(t, i), ofp ^t –1–i, with 1 t m and 0 i < p–1. We then study such privileged codes. We first make precize which duals of extendedBCH codes they are. Next, we show that Weil's bound in this context gives an explicit formula; that is, the couple (t, i) fully determines the value of the Weil bound for the code with borderF(t, i). In the case where this value is negative, we use the Roos method to bound the minimum distance, greatly improving the BCH bound.     

On a signal processing algorithms based class of linear codes

In this work, the correspondence between linear (n,k,d) codes and aperiodic convolution algorithms for computing a system ofk bilinear forms over GF(p^m) is explored. A number of properties are established for the linear codes that can be obtained from a computational procedure of this type. A particular bilinear form is considered and a class of linear codes over GF(2^m) is derived with varyingk andd parameters. The code lengthn is equal to the multiplicative complexity of the computation of an aperiodic convolution and an efficient computation thereof leads to the shortest codes possible using this approach, many of which are optimal or near-optimal. A new decoding procedure for this class of linear codes is presented which exploits the block structure of the generator matrix of the codes. Several interesting observations are made on the nature of the codes obtained as a result of such computations. Such a computation of bilinear forms can be generalized to include other bilinear forms and the related classes of codes.     

Structural properties and enumeration of quasi cyclic codes

Given any finite fieldF _q , an (N, K) quasi cyclic code is defined as aK dimensional linear subspace ofF _q ^N which is invariant underT ⁿ for some integern, 0 <n N, and whereT is the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to theF _q []-submodules ofF _q ^N where the product(gl)· is naturally defined as ₀ + ₁T ⁿ +...+ _m T ^mn if()= ₀ + ₁ +...+ _{m m} .In the case where (N/n, q)=1, all quasi cyclic codes are shown to be decomposable into the direct sum of a fixed number of indecomposable components called irreducible cyclicF _q []-submodules providing for the complete characterisation and enumeration of some subclasses of quasi cyclic codes including the cyclic codes, the quasi cyclic codes with a cyclic basis, the maximal and the irreducible ones. Finally a general procedure is presented which allows for the determination and characterisation of the dual of any quasi cyclic code.     

Transform domain characterization of cyclic codes overZ m

Cyclic codes with symbols from a residue class integer ringZ _m are characterized in terms of the discrete Fourier transform (DFT) of codewords defined over an appropriate extension ring ofZ _m . It is shown that a cyclic code of length n overZ _m ,n relatively prime tom, consists ofn-tuples overZ _m having a specified set of DFT coefficients from the elements of an ideal of a subring of the extension ring. Whenm is equal to a product of distinct primes every cyclic code overZ _m has an idempotent generator and it is shown that the idempotent generators can be easily identified in the transform domain. The dual code pairs overZ _m are characterized in the transform domain for cyclic codes. Necessary and sufficient conditions for the existence of self-dual codes overZ _m are obtained and nonexistence of self-dual codes for certain values ofm is proved.     

On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form $y^q+y=x^m,\,q$ being a prime power and $m$ a positive integer which divides $q+1$ . The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.     

Complex zeros of a transcendental impedance function

Summary The electrical impedance of a semiconductor supporting two waves contains an entire transcendental function of the form f(z)=exp(–z)–1–cz, wherec is a complex parameter. This function has an infinity of zeros in the left half z-plane when c is finite (0<¦c¦<+). Several approximate expressions for the location of zeros as function of c are obtained. For certain values of c (cf. Fig. 3) one or several zeros are located in the right half z-plane. The real part of some of those exceeds an arbitrarily large positive number, provided c is properly chosen. This corresponds to resonances which represent growing oscillations.     

Algorithm for a class of constrained weight least square problems and applications

An algorithm, based on ordinal optimisation (OO) and sensitive theories, is presented to solve a class of constrained weight least square problems with continuous and discrete variables. the proposed algorithm can cope with an enormous amount of computational complexity problems and has a high probability of obtaining a good enough solution according to the oo theory. this method has some advantages, such as computational efficiency, numerical stability and the superiority of the good enough solution. the proposed algorithm is explicit, compact and easy to program. test results demonstrate that the proposed approach is more computational-efficient than other existing approaches for solving constrained-state estimation problems with continuous and discrete variables on the ieee 30-bus and the ieee 118-bus systems.     

Polynomial-time construction of codes I: Linear codes with almost equal weights

We prove that there are infinite families (C_i)_i0 of codes over F_q with polynomial complexity of construction whose relative weights are as close to as we want and are such that

Improved CUSUM monitoring of Markov counting process with frequent zeros

There has been a growing interest in monitoring processes featuring serial dependence and zero inflation. The phenomenon of excessive zeros often occurs in count time series because of the advancement of quality in manufacturing process. In this study, we propose three control charts, such as the cumulative sum chart with delay rule (CUSUM‐DR), conforming run length (CRL)‐CUSUM chart, and combined Shewhart CRL‐CUSUM chart, to enhance the performance of monitoring Markov counting processes with excessive zeros. Numerical experiments are conducted based on integer‐valued autoregressive time series models, for example, zero‐inflated Poisson INAR and INARCH, to evaluate the performance of the proposed charts designed for the detection of mean increase. A real example is also illustrated to demonstrate the usability of our proposed charts.     

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the ‘hypercube’ polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the ‘simplex’ polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.     

The negative binomial-Lindley distribution as a tool for analyzing crash data characterized by a large amount of zeros

The modeling of crash count data is a very important topic in highway safety. As documented in the literature, given the characteristics associated with crash data, transportation safety analysts have proposed a significant number of analysis tools, statistical methods and models for analyzing such data. Among the data issues, we find the one related to crash data which have a large amount of zeros and a long or heavy tail. It has been found that using this kind of dataset could lead to erroneous results or conclusions if the wrong statistical tools or methods are used. Thus, the purpose of this paper is to introduce a new distribution, known as the negative binomial–Lindley (NB-L), which has very recently been introduced for analyzing data characterized by a large number of zeros. The NB–L offers the advantage of being able to handle this kind of datasets, while still maintaining similar characteristics as the traditional negative binomial (NB). In other words, the NB–L is a two-parameter distribution and the long-term mean is never equal to zero. To examine this distribution, simulated and observed data were used. The results show that the NB–L can provide a better statistical fit than the traditional NB for datasets that contain a large amount of zeros.     

Synthesis of the low-temperature co-fired ceramic bandpass filters and diplexer with transmission zeros

The multilayered microwave filters are developed with the assistance of two transmission zeros. Besides a coupled line connected to one capacitor, another capacitor cross coupled between the source and load ports is utilised for the proposed filters. By properly adjusting the coupling coefficients between the coupled line and two capacitors, two transmission zeros would appear on the lower side, the higher side or on both sides of the passband skirt. Thus, the characteristics can be employed for diplexers or dual-band filters to make dual-mode portable telephones. Moreover, to miniaturise the circuit sizes and improve their performances, multilayered structure and the low-temperature co-fired ceramic technology are adopted for filter fabrication. The theorem of filter synthesis and procedures of filter design are also presented. Finally, the proposed structure is validated by an agreement between measured results of fabricated filters and the electromagnetic simulation.     

Design of low-temperature co-fired ceramic bandpass filter with transmission zeros in the stopband

A multilayered bandpass filter with low-temperature co-fired ceramic technology is proposed. By integrating two types of filters as one filter, four transmission zeros can be generated. Each type of filter with a source-load coupling capacitor and a grounded coupling inductor, simultaneously, can generate two transmission zeros located on the opposite side of the passband skirt separately. Moreover, the fabricated filter provides good matching with the electromagnetic simulation and can prove the availability of the filter configuration.     

Dynamics of a nonlinear periodic structure with cyclic symmetry

Summary The free and forced motions of a nonlinear periodic structure with cyclic symmetry are studied. The structure consists of a number of identical linear flexural members coupled by means of nonlinear stiffnesses of the third degree. It is found that this system can only possessn similar nonlinear modes of free oscillation, and that no other modes are possible. Moreover, there exist pairs of nonlinear modes with mutually orthogonal nodal diameters having, in general, distinct backbone curves. A multiple-scales averaging analysis is used to study the nonlinear interaction between a pair of modes with orthogonal nodal diameters. As a result of this analysis, it is found that all pairs of nonliner modes along with all their linear combinations are orbitally unstable, and the only possible orbitally stable periodic motions are free travelling waves, that propagate through the structure in the clockwise and anti-clockwise directions. Under harmonic forcing, a bifuraction of a stable branch of forced travelling waves from a branch of forced normal mode motions is detected, and jump phenomena between branches of periodic solutions are observed. The analytical results are in agreement with experimental observations of an earlier work, and, in addition, are verified by numerical simulations.     

Dispersion relation for real-plane zeros as a concept of wave-front measurement

The functional relationship between the phase logarithm and the amplitude logarithm of a wave function near its real-plane zero point is found. This result takes the form of the dispersion relation that is deduced analytically and supported by the numerical simulation of the light-wave propagation in an inhomogeneous medium. The sufficient and necessary conditions of existence of this relationship are discussed, and their validity for infinite spectra is shown.