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

      Dougherty, Steven; Korban, Adrian; Șahinkaya, Serap; Ustun, Deniz (American Institute of Mathematical Sciences (AIMS), 2022)
      <p style='text-indent:20px;'>We present a generator matrix of the form <inline-formula><tex-math id="M1">\begin{document}$ [ \sigma(v_1) \ | \ \sigma(v_2)] $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ v_1 \in RG $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ v_2\in RH $\end{document}</tex-math></inline-formula>, for finite groups <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> of order <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula> for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve <inline-formula><tex-math id="M8">\begin{document}$ 10 $\end{document}</tex-math></inline-formula> of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain <inline-formula><tex-math id="M9">\begin{document}$ 82 $\end{document}</tex-math></inline-formula> new binary linear complementary dual codes, <inline-formula><tex-math id="M10">\begin{document}$ 50 $\end{document}</tex-math></inline-formula> of which are either optimal or near optimal of lengths <inline-formula><tex-math id="M11">\begin{document}$ 41 \leq n \leq 61 $\end{document}</tex-math></inline-formula> which are new to the literature.</p>
    • DNA codes from skew dihedral group ring

      Dougherty, Steven; Korban, Adrian; Şahinkaya, Serap; Ustun, Deniz (American Institute of Mathematical Sciences (AIMS), 2022)
      <p style='text-indent: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 <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{F}_{j, k} $\end{document}</tex-math></inline-formula> and its associated Gray maps, we show how one can construct reversible codes of length <inline-formula><tex-math id="M2">\begin{document}$ n2^{j+k} $\end{document}</tex-math></inline-formula> over the finite field <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_4. $\end{document}</tex-math></inline-formula> 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.</p>
    • The multi-dimensional Stochastic Stefan Financial Model for a portfolio of assets

      Antonopoulou, Dimitra; Bitsaki, Marina; Karali, Georgia; University of Chester; University of Crete (American Institute of Mathematical Sciences (AIMS), 2021-04-01)
      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.