Browsing Faculty of Science and Engineering by Publisher "American Institute of Mathematical Sciences (AIMS)"
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Binary selfdual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme<p style='textindent:20px;'>We present a generator matrix of the form <inlineformula><texmath id="M1">\begin{document}$ [ \sigma(v_1) \  \ \sigma(v_2)] $\end{document}</texmath></inlineformula>, where <inlineformula><texmath id="M2">\begin{document}$ v_1 \in RG $\end{document}</texmath></inlineformula> and <inlineformula><texmath id="M3">\begin{document}$ v_2\in RH $\end{document}</texmath></inlineformula>, for finite groups <inlineformula><texmath id="M4">\begin{document}$ G $\end{document}</texmath></inlineformula> and <inlineformula><texmath id="M5">\begin{document}$ H $\end{document}</texmath></inlineformula> of order <inlineformula><texmath id="M6">\begin{document}$ n $\end{document}</texmath></inlineformula> for constructing selfdual codes and linear complementary dual codes over the finite Frobenius ring <inlineformula><texmath id="M7">\begin{document}$ R $\end{document}</texmath></inlineformula>. In general, many of the constructions to produce selfdual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find selfdual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve <inlineformula><texmath id="M8">\begin{document}$ 10 $\end{document}</texmath></inlineformula> of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain <inlineformula><texmath id="M9">\begin{document}$ 82 $\end{document}</texmath></inlineformula> new binary linear complementary dual codes, <inlineformula><texmath id="M10">\begin{document}$ 50 $\end{document}</texmath></inlineformula> of which are either optimal or near optimal of lengths <inlineformula><texmath id="M11">\begin{document}$ 41 \leq n \leq 61 $\end{document}</texmath></inlineformula> which are new to the literature.</p>

DNA codes from skew dihedral group ring<p style='textindent:20px;'>In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring <inlineformula><texmath id="M1">\begin{document}$ \mathcal{F}_{j, k} $\end{document}</texmath></inlineformula> and its associated Gray maps, we show how one can construct reversible codes of length <inlineformula><texmath id="M2">\begin{document}$ n2^{j+k} $\end{document}</texmath></inlineformula> over the finite field <inlineformula><texmath id="M3">\begin{document}$ \mathbb{F}_4. $\end{document}</texmath></inlineformula> As an application, we construct a number of DNA codes that satisfy the Hamming distance, the reverse, the reversecomplement, and the GCcontent constraints with better parameters than some good DNA codes in the literature.</p>

The multidimensional Stochastic Stefan Financial Model for a portfolio of assetsThe financial model proposed in this work involves the liquidation process of a portfolio of n assets through sell or (and) buy orders placed, in a logarithmic scale, at a (vectorial) price with volatility. We present the rigorous mathematical formulation of this model in a financial setting resulting to an ndimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading, the socalled solid phase. We will focus on a case of financial interest when one or more markets are considered. In particular, our aim is to estimate for a short time period the areas of zero trading, and their diameter which approximates the minimum of the n spreads of the portfolio assets for orders from the n limit order books of each asset respectively. In dimensions n = 3, and for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by Niethammer in [25], and in [7] in a more general setting. There in, when the initial moving boundary consists of well separated spheres, a first order approximation system of odes had been rigorously derived for the dynamics of the interfaces and the asymptotic pro le of the solution. In our financial case, we propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices, while each sphere may correspond to a different market; the relevant radii represent the half of the minimum spread. We apply It^o calculus and provide second order formal asymptotics for the stochastic version dynamics, written as a system of stochastic differential equations for the radii evolution in time. A second order approximation seems to disconnect the financial model from the large diffusion assumption for the trading density. Moreover, we solve the approximating systems numerically.