Now showing items 1-3 of 3

• #### Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme

&lt;p style='text-indent:20px;'&gt;We present a generator matrix of the form &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$[ \sigma(v_1) \ | \ \sigma(v_2)]$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, where &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$v_1 \in RG$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$v_2\in RH$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, for finite groups &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$G$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$H$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of order &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$n$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$R$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. In general, many of the constructions to produce self-dual 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 self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$10$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; 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 &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$82$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; new binary linear complementary dual codes, &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$50$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of which are either optimal or near optimal of lengths &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$41 \leq n \leq 61$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; which are new to the literature.&lt;/p&gt;
• #### DNA codes from skew dihedral group ring

&lt;p style='text-indent:20px;'&gt;In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$\mathcal{F}_{j, k}$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and its associated Gray maps, we show how one can construct reversible codes of length &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$n2^{j+k}$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; over the finite field &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$\mathbb{F}_4.$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; As an application, we construct a number of DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement, and the GC-content constraints with better parameters than some good DNA codes in the literature.&lt;/p&gt;
• #### The multi-dimensional Stochastic Stefan Financial Model for a portfolio of assets

The 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 n-dimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading, the so-called 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.