The dual of a constacyclic code,self dual,reversible constacyclic codes and constacyclic codes with complementary duals over finite local Frobenius non-chain rings with nilpotency index 3

Over finite local Frobenius non-chain rings with nilpotency index 3 and when the length of the codes is relatively prime to the characteristic of the residue field of the ring, the structure of the dual of $\gamma$-constacyclic codes is established and the algebraic characterization of self-dual $\gamma$-constacyclic codes, reversible $\gamma$-constacyclic codes and $\gamma$-constacyclic codes with complementary dual are given. Generators for the dual code are obtained from those of the original constacyclic code.     

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.     

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.