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.     

Long binary narrow-sense BCH codes are normal

Let C be the binary narrow-sense BCH code of length n = (2 ^m − l)/h, where m is the order of 2 modulo n. Using characters of finite fields and a theorem of Weil, and results of Vladut-Skorobogatov and Lang-Weil we prove that the code C is normal in the non-primitive case h > 1 if 2^m ≥ 4(2th)^{4t + 2}, and in the primitive case h = 1 if m ≥ m ₀ where the constant m ₀ depends only on t.     

Higher weights for codes over rings

A generalized definition of higher weights for codes over finite chain rings and principal ideal rings and bounds on the minimum higher weights in this setting are given. Using this we generalize the definition for higher MDS and MDR codes. Computationally, the higher weight enumerator of lifted Hamming and Simplex codes over \mathbbZ₄{\mathbb{Z}_4}, the minimum higher weights for the lifted code of the binary [8,4,4] self-dual extended Hamming code, the lifted code of the ternary [12,6,6] self-dual Golay code and the lifted code of the binary [24,12,8] self-dual Golay code are given. Joint weight enumerators are used to produce MacWilliams relations for specific higher weight enumerators.     

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.     

Primitive Polynomials over Finite Fields of Characteristic Two

Let F _q be a finite field of characteristic two. We prove that for any given element a F _q , there exists a primitive polynomial of degree n over F _q with the m-th (0<m<n) coefficient _m =a when n7, n is odd. Keywords:Finite field, Primitive polynomial, Galois ring, Character sums over Galois ring, The sieve method.     

A note on the moments of Kloosterman sums

It is well known that Kloosterman sums have many important applications in many subjects, such as finding the roots of an equation over finite fields or the rational points on an algebraic curve, determining the weight distributions of some algebraic geometric codes, calculating some exponential sums in number theory etc. We provide some new identities and results on the moments of Kloosterman sums.     

On bilevel machine scheduling problems

Bilevel scheduling problems constitute a hardly studied area of scheduling theory. In this paper, we summarise the basic concepts of bilevel optimisation, and discuss two problem classes for which we establish various complexity and algorithmic results. The first one is the bilevel total weighted completion time problem in which the leader assigns the jobs to parallel machines and the follower sequences the jobs assigned to each machine. Both the leader and the follower aims to minimise the total weighted completion time objective, but with different job weights. When the leader’s weights are arbitrary, the problem is NP-hard. However, when all the jobs are of unit weight for the leader, we provide a heuristic algorithm based on iterative LP-rounding along with computational results, and provide a sufficient condition when the LP-solution is integral. In addition, if the follower weights induce a monotone (increasing or decreasing) processing time order in any optimal solution, the problem becomes polynomially solvable. As a by-product, we characterise a new polynomially solvable special case of the MAX m-CUT problem, and provide a new linear programming formulation for the P||?_j C_j{P||\sum_j C_j} problem. Finally, we present some results on the bilevel order acceptance problem, where the leader decides on the acceptance of orders and the follower sequences the jobs. Each job has a deadline and if a job is accepted, it cannot be late. The leader’s objective is to maximise the total weight of accepted jobs, whereas the follower aims at minimising the total weighted job completion times. For this problem, we generalise some known single-level machine scheduling algorithms.     

On the summation of certain legendre series

Summary The purpose of this paper is to find some recurrence relations for sums of infinite series of the form P _l ⁿ (cos)t ^l+m/(l+m). This is achieved by transforming the sums into integrals and then using the recurrence relations for these integrals.On leave from the Department of Mathematics, Kurukshetra University, Kurukshetra, India.     

Non-degenerate Linear Codes Determined by their Weight Enumerators

 In [1], Homma-Kim-Yoo showed that a code C such that the support of the associated projective system is a union of two linear subspaces, is uniquely determined up to equivalence by its weight enumerator. Let W _C (z) be the weight enumerator of a code C. Assuming certain conditions on the coefficients of W _C (z) for some range, we prove that the support of the associated projective system is a union of two linear subspaces. Received: February 27, 2002; revised version: October 2, 2002 Keywords: Linear code, Projective system, Weight enumerator. MSC2000: 94B27, 94B65, 51E20 Partially supported by Grant-in-Aid for Scientific Research, Ministry of Education, #12640180     

Selecting the particle size distribution for drugs with low water solubility – mathematical model

Purpose: To introduce guidelines in selecting the particle size distribution (n₀, cm^?1) that will guarantee optimal oral absorption for drugs with low solubility.

Methods: Unlike other multi-compartmental models the gastrointestinal tract is modeled as a continuous tube with spatially varying properties. The transport through the intestinal lumen is described using the dispersion model. The model accounts for the dissolution of poly-dispersed powders.

Results: The model was used to examine the sensitivity of the absorption on permeability (P) and water solubility (C_s) following administration in different log-normal powders. The absorption exhibits inverse sigmoidal dependence on the mean particle size (r_m, µm) regardless of the administrated dose or drug properties. Thus, there is an optimal r_m that maximizes the benefit-cost ratio of the formulation; finer particles do not improve the absorption while coarser particles decrease it. Using the model we find that the optimal r_m depends mainly on the drug C_s and on the geometrical standard deviation (gSTD).

Conclusions: The results of this work provide the formulator with guidelines to select both r_m and gSTD that guarantee optimal absorption.     

The Importance of Phase Information for Constructing Model Partially Coherent Sources

Abstract

An alternative factorisation of the mutual coherence function is presented, given by Γ(r ₁, r ₂, τ) = V(r ₁)V?(r ₂)η(r ₁, r ₂, τ), where V(r) is the time independent part of the coherent analytic signal and η (r ₁, r ₂, τ) is a coherence function. We show how simple coherent model sources, which are composed of sums of plane waves, can be converted into partially coherent model sources. As examples of this we give the most general expressions for the uniform intensity and sinusoidal intensity sources of infinite extent. Modal power coupling calculations also shown to be significantly simplified, for sources with simple sums of plane waves in their coherent limit.     

Constraints on Weights in Binary Codes

 There are known constraints on the number of doubly-even and singly-even vectors in a linear even binary code C. These constraints give information about C∩C ^⊥. We find new constraints on weights in a binary linear code C which also contains odd weight vectors. This leads to information about the dimension and weights of C∩C ^⊥. Received: September 20, 1996     

Light-Responsive Colloidal Crystals Engineered with DNA

A novel method for synthesizing and photopatterning colloidal crystals via light-responsive DNA is developed. These crystals are composed of 10–30 nm gold nanoparticles interconnected with azobenzene-modified DNA strands. The photoisomerization of the azobenzene molecules leads to reversible assembly and disassembly of the base-centered cubic (bcc) and face-centered cubic (fcc) crystalline nanoparticle lattices. In addition, UV light is used as a trigger to selectively remove nanoparticles on centimeter-scale thin films of colloidal crystals, allowing them to be photopatterned into preconceived shapes. The design of the azobenzene-modified linking DNA is critical and involves complementary strands, with azobenzene moieties deliberately staggered between the bases that define the complementary code. This results in a tunable wavelength-dependent melting temperature (T_m) window (4.5–15 °C) and one suitable for affecting the desired transformations. In addition to the isomeric state of the azobenzene groups, the size of the particles can be used to modulate the T_m window over which these structures are light-responsive.     

Reliability approximation using finite Weibull mixture distributions

The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that the reliability of an arbitrary system can be approximated well by a finite Weibull mixture with positive component weights only, without knowing the structure of the system, on condition that the unknown parameters of the mixture can be estimated. To support the main idea, five examples are presented. In order to estimate the unknown component parameters and the component weights of the Weibull mixture, some of the already existing methods are applied and the EM algorithm for the m-fold Weibull mixture is derived. The fitted distributions obtained by different methods are compared to the empirical ones by calculating the AIC and δ_C values. It can be concluded that the suggested Weibull mixture with an arbitrary but finite number of components is suitable for lifetime data approximation. For parameter estimation the combination of the alternative and EM algorithm is suggested.     

Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes

 For any integral convex polytope in ℝ there is an explicit construction of an error-correcting code of length (q-1)² over the finite field 𝔽 _q , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is determined using the cohomology and intersection theory of the underlying surfaces. In detail we treat Hirzebruch surfaces. Received: August 21, 2000; revised version: September 3, 2002     

Comparative study of size and distribution of lamellar thicknesses and long periods in polyethylene with a shish-kebab structure

This study contains a combined application of three different techniques for the study of injection moulded polyethylene (PE), showing an oriented shish-kebab structure: small angle X-ray scattering (SAXS), low frequency Raman spectroscopy (LAM) and transmission electron microscopy (TEM), A series of linear PEs and molecular weights in the range 51000–478000 has been investigated and two injection temperatures have been used (T _m=144 and 210 °C). SAXS patterns from the highly oriented regions show the presence of either one axial long period (L ₁) or two (L ₁ and L ₂) depending on molecular weight (¯M _w) and T _m. It is shown that L ₁ and L ₂ increase with ¯M _w up to a given critical molecular weight ¯M _c. Above ¯M _c, L ₁ and L ₂ remain constant. Raman results qualitatively confirm the existence of two separate distributions of straight-length chain segments for those samples having molecular weights above the critical value. Shorter segments are shown to be more abundant than the longer ones. In the lowest molecular weight sample, results from SAXS, TEM and Raman spectroscopy seem to be consistent with each other, although in some cases a tilted molecular arrangement within the lamellae has to be invoked. On the other hand, in case of the highest molecular weight sample, the length of the short straight-chain segments derived from Raman spectroscopy agrees well with the double periodicity obtained from SAXS. On the contrary, long periods measured from TEM only correspond to the shorter SAXS periodicity. This result is discussed by assuming the occurrence of crystalline bridges among adjacent lamellae.     

The number of MDS [7, 3] codes on finite fields of characteristic 2

A code of lengthn, dimensionk and minimum distanced ismaximum distance separable (MDS) ifk+d=n+1. We give the number of MDS codes of length 7 and dimension 3 on finite fields withq elements whereq=2 ^m . In order to get this number, we compute the number of configurations of seven points in the projective plane overF _q , no three of which are collinear.     

Brucine-loaded liposomes composed of HSPC and DPPC at different ratios: in vitro and in vivo evaluation

Objective: The objective of this study is to test the hypothesis that the phase transition temperature (T_m), the main property of liposomes, can be easily controlled by changing the molar ratio of hydrogenated soy phosphatidylcholine (HSPC) and 1,2-dipalmitoyl-sn-glycero-3-phosphacholine (DPPC) after drug encapsulation.

Materials and methods: Brucine, an antitumor alkaloid, was encapsulated into the liposomes with different HSPC/DPPC compositions. The T_ms of the brucine-loaded liposomes (BLs) were determined by differential scanning calorimetry (DSC). Then the physicochemical properties and pharmacokinetics of the BLs with different HSPC/DPPC compositions were investigated and compared.

Results: The results of DSC revealed that HSPC and DPPC can combine into one phase. The findings of molecular modeling study suggested that HSPC interacts with DPPC via electrostatic interaction. The molar ratio of HSPC/DPPC influenced the sizes of BLs but had little effect on the entrapment efficiency (EE). The stability of BLs was improved with the increase of the HSPC ratios, especially with the presence of plasma. Following i.v. administration, it was found that AUC values of BLs in vivo were directly related to the HSPC/DPPC ratios of BLs, namely the T_ms of BLs.

Discussion: The behavior of liposomes, especially in vivo pharmacokinetic behavior, can be controlled by the modification of T_m.

Conclusion: The characterization of BLs in vitro and in vivo had demonstrated that the T_m could be flexibly modified for liposomes composed of both HSPC and DPPC. Using HSPC/DPPC composition may be an efficient strategy to control the T_m, thus control the in vivo pharmacokinetic behavio, of BLs.     

1-Perfect Uniform and Distance Invariant Partitions

Let F ⁿ be the n-dimensional vector space over ℤ₂. A (binary) 1-perfect partition of F ⁿ is a partition of F ⁿ into (binary) perfect single error-correcting codes or 1-perfect codes. We define two metric properties for 1-perfect partitions: uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation invariant and not translation invariant propelinear codes. The search for examples of 1-perfect uniform but not distance invariant partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7. Received: March 6, 2000; revised version: November 30, 2000     

Optimal Formally Self-Dual Codes over ?5 and ?7

In this paper, we study optimal formally self-dual codes over ?₅ and ?₇. We determine the highest possible minimum weight for such codes up to length 24. We also construct formally self-dual codes with highest minimum weight, some of which have the highest minimum weight among all known linear codes of corresponding length and dimension. In particular, the first known [14, 7, 7] code over ?₇ is presented. We show that there exist formally self-dual codes which have higher minimum weights than any comparable self-dual codes. Received: May 18, 1998; revised version: September 4, 1999