Weight enumerators of reducible cyclic codes and their dual codes

Let ${\mathbb{F}}_q$ be a finite field and $n$ a positive integer. In this paper, we find a new combinatorial method to determine weight enumerators of reducible cyclic codes and their dual codes of length $n$ over ${\mathbb{F}}_q$, which just generalize results of Zhu et al. (2015); especially, we also give the weight enumerator of a cyclic code, which is viewed as a partial Melas code. Furthermore, weight enumerators obtained in this paper are all in the form of power of a polynomial.     

The support designs of the triply even binary codes of length 48

In this paper, we present examples of codes all of whose weight classes support 1‐designs, with duals whose classes include two that support 2‐designs. We can find these examples in the triply even binary codes of length 48, which have been classified by Betsumiya and Munemasa.     

An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

The problem of interpolation of scattered data on the unit sphere has many applications in geodesy and Earth science in which the sphere is taken as a model for the Earth. Spherical radial basis functions provide a convenient tool for constructing the interpolant. However, the underlying linear systems tend to be ill-conditioned. In this paper, we present an additive Schwarz preconditioner for accelerating the solution process. An estimate for the condition number of the preconditioned system will be discussed. Numerical experiments using MAGSAT satellite data will be presented.     

Alkane isomers: presence in petroleum ether and complexity

The presence and absence of alkane isomers in petroleum and petroleum derivatives depend on the complexity of these structures. It was assumed that the more complex the structure is the less probable it is that that the molecule can be detected in any petroleum derivative. Complexity is a vague concept, which has not been defined in quantitative terms yet, and therefore there is no experimental method, which could be used to determine ‘complexity’. Mass spectrometry and infrared spectroscopy in combination with gas chromatography were used to identify the various structural isomers of alkanes in petroleum ether. The isomers were categorised in quantitative terms by using topological indices and linear discriminant analysis. It was found that alkanes possessing a more complex, highly branched structure are less probable to be detected in petroleum ether than isomers with a simpler backbone structure. It was proposed that the experimental ‘measure’ of the complexity of isomer_i should be proportional to 1/C_i , where C_i , denotes the concentration of isomer_i in a (primary) petroleum derivative.