Some Upper Bounds on the Covering Radii of Linear Codes Over Fq and Their Applications

We show that the covering radius R of an [n,k,d] code over F_q is bounded above by R n-n _q(k, d/q). We strengthen this bound when R d and find conditions under which equality holds.As applications of this and other bounds, we show that all binary linear codes of lengths up to 15, or codimension up to 9, are normal. We also establish the normality of most codes of length 16 and many of codimension 10. These results have applications in the construction of codes that attain t[n,k,/it>], the smallest covering radius of any binary linear [n,k].We also prove some new results on the amalgamated direct sum (ADS) construction of Graham and Sloane. We find new conditions assuring normality of the ADS; covering radius 1 less than previously guaranteed for ADS of codes with even norms; good covering codes as ADS without the hypothesis of normality, from concepts p- stable and s- stable; codes with best known covering radii as ADS of two, often cyclic, codes (thus retaining structure so as to be suitable for practical applications).     

Tandem Ring Opening and Oximation of Ethyl 3‐Aroyl‐1‐cyano‐4‐hydroxy‐2,4,6‐triarylcyclohexanecarboxylate by Hydroxylamine

The reaction of hydroxylamine hydrochloride with ethyl 3‐aroyl‐1‐cyano‐4‐hydroxy‐2,4,6‐triarylcyclohexanecarboxylate has led to the formation of ethyl 2‐[amino(hydroxyimino)methyl]‐3‐aryl‐5‐(hydroxyimino)‐5‐arylpentanoate via a tandem ring opening and oximation process. The structures of the products are confirmed by NMR and X‐ray techniques, and the conformational features of the product arrived at are compared with a molecular dynamics simulation study.     

Eigenvalues and expansion of bipartite graphs

We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of connected bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes of graphs that are optimal with respect to the bounds. We prove that BIBD-graphs are characterized by their eigenvalues. Finally we present a new bound on the expansion coefficient of (c, d)-regular bipartite graphs and compare that with with a classical bound.     

McEliece Public Key Cryptosystems Using Algebraic-Geometric Codes

McEliece proposed a public-key cryptosystem based on algebraic codes, in particular binary classical Goppa codes. Actually, his scheme needs only a class of codes with a good decoding algorithm and with a huge number of inequivalent members with given parameters. In the present paper we look at various aspects of McEliece's scheme using the new and much larger class of R-ary algebraic-geometric Goppa codes.     

On the conjecture on APN functions and absolute irreducibility of polynomials

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field \(\mathbb {F}\) is called exceptional APN, if it is also APN on infinitely many extensions of \(\mathbb {F}\). In this article we consider the most studied case of \(\mathbb {F}=\mathbb {F}_{2^n}\). A conjecture of Janwa–Wilson and McGuire–Janwa–Wilson (1993/1996), settled in 2011, was that the only monomial exceptional APN functions are the monomials \(x^n\), where \(n=2^k+1\) or \(n={2^{2k}-2^k+1} \) (the Gold or the Kasami exponents, respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our results is that all functions of the form \(f(x)=x^{2^k+1}+h(x)\) (for any odd degree h(x), with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture. We also show absolute irreducibility of a class of multivariate polynomials over finite fields (by repeated hyperplane sections, linear transformations, and reductions) and discuss their applications.     

On the subfield subcodes of Hermitian codes

We present a fast algorithm using Gröbner basis to compute the dimensions of subfield subcodes of Hermitian codes. With these algorithms we are able to compute the exact values of the dimension of all subfield subcodes up to q ≤  32 and length up to 2¹⁵. We show that some of the subfield subcodes of Hermitian codes are at least as good as the previously known codes, and we show the existence of good long codes.     

Superhydrophobic surfaces with low and high adhesion made from mixed (hydrocarbon and fluorocarbon) 3,4‐propylenedioxythiophene monomers

This work concerns new superhydrophobic surfaces, generated by replacing long fluorocarbon chains, which bioaccumulate, with short chains whilst at the same time retaining oleophobic properties. Here, is described the synthesis of novel 3,4‐propylenedioxythiophene derivatives containing both a short fluorocarbon chain (perfluorobutyl) and a hydrocarbon chain of various lengths (ethyl, butyl and hexyl). Superhydrophobic (θ_water > 150°) surfaces with good oleophobic properties (60° > θ_hexadecane > 80°) have been obtained by electrodeposition using cyclic voltammetry. Surprisingly, the lowest hystereses and sliding angles (Lotus effect) are obtained with the shortest alkyl chains due to the presence of microstructures made of nanofibers on the surfaces, whereas, the longest alkyl chains leads to nanosheets with high adhesion (Petal effect). Such materials are potential candidates for biomedical applications. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 782–788