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New Selfdual Codes from 2 x 2 block circulant matrices, Group Rings and Neighbours of NeighboursIn this paper, we construct new selfdual codes from a construction that involves a unique combination; $2 \times 2$ block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields selfdual codes. The theory is supported by the construction of selfdual codes over the rings $\FF_2$, $\FF_2+u\FF_2$ and $\FF_4+u\FF_4$. Using extensions and neighbours of codes, we construct $32$ new selfdual codes of length $68$. We construct 48 new best known singlyeven selfdual codes of length 96.

Layer Dynamics for the one dimensional $\eps$dependent CahnHilliard / AllenCahn EquationWe study the dynamics of the onedimensional εdependent CahnHilliard / AllenCahn equation within a neighborhood of an equilibrium of N transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a massconservation constraint in place of one of the zeromass flux boundary conditions at x = 1. Motivated by the study of Carr and Pego on the layered metastable patterns of AllenCahn in [10], and by this of Bates and Xun in [5] for the CahnHilliard equation, we implement an Ndimensional, and a massconservative N−1dimensional manifold respectively; therein, a metastable state with N transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized CahnHilliard / AllenCahn operator, and specify wide families of εdependent weights δ(ε), µ(ε), acting at each part of the operator, for which the dynamics are stable and rest exponentially small in ε. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to AllenCahn, or, when the mass is conserved, close to the CahnHilliard solution.

New Extremal Binary Selfdual Codes from block circulant matrices and block quadratic residue circulant matricesIn this paper, we construct selfdual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield selfdual codes. We construct selfdual codes of various lengths over F2 and F2 + uF2. Using extensions, neighbours and sequences of neighbours, we construct many new selfdual codes. In particular, we construct one new selfdual code of length 66 and 51 new selfdual codes of length 68.

Spatial discretization for stochastic semilinear subdiffusion driven by integrated multiplicative spacetime white noiseSpatial discretization of the stochastic semilinear subdiffusion driven by integrated multiplicative spacetime white noise is considered. The spatial discretization scheme discussed in Gy\"ongy \cite{gyo_space} and Anton et al. \cite{antcohque} for stochastic quasilinear parabolic partial differential equations driven by multiplicative spacetime noise is extended to the stochastic subdiffusion. The nonlinear terms $f$ and $\sigma$ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the integrated multiplicative spacetime white noise are discretized by using finite difference methods. Based on the approximations of the Green functions which are expressed with the MittagLeffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under the suitable smoothness assumptions of the initial values.

Oscillatory and stability of a mixed type difference equation with variable coefficientsThe goal of this paper is to study the oscillatory and stability of the mixed type difference equation with variable coefficients \[ \Delta x(n)=\sum_{i=1}^{\ell}p_{i}(n)x(\tau_{i}(n))+\sum_{j=1}^{m}q_{j}(n)x(\sigma_{i}(n)),\quad n\ge n_{0}, \] where $\tau_{i}(n)$ is the delay term and $\sigma_{j}(n)$ is the advance term and they are positive real sequences for $i=1,\cdots,l$ and $j=1,\cdots,m$, respectively, and $p_{i}(n)$ and $q_{j}(n)$ are real functions. This paper generalise some known results and the examples illustrate the results.

Error estimates of a continuous Galerkin time stepping method for subdiffusion problemA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \, \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and Ltype methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

New binary selfdual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction.In this work, we give a new technique for constructing selfdual codes over commutative Frobenius rings using $\lambda$circulant matrices. The new construction was derived as a modification of the wellknown four circulant construction of selfdual codes. Applying this technique together with the buildingup construction, we construct singlyeven binary selfdual codes of lengths 56, 58, 64, 80 and 92 that were not known in the literature before. Singlyeven selfdual codes of length 80 with $\beta \in \{2,4,5,6,8\}$ in their weight enumerators are constructed for the first time in the literature.

Composite Matrices from Group Rings, Composite GCodes and Constructions of SelfDual CodesIn this work, we define composite matrices which are derived from group rings. We extend the idea of Gcodes to composite Gcodes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite Gcode is also a composite Gcode. We also define quasicomposite Gcodes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary selfdual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.

GCodes, selfdual GCodes and reversible GCodes over the Ring Bj,kIn this work, we study a new family of rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study Gcodes, selfdual Gcodes, and reversible Gcodes over this family. In particular, we show that the projection of a Gcode over Bj,k to a code over Bl,m is also a Gcode and the image under the Gray map of a selfdual Gcode is also a selfdual Gcode when the characteristic of the base field is 2. Moreover, we show that the image of a reversible Gcode under the Gray map is also a reversible G2j+kcode. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasiG codes, which are the images of Gcodes under the Gray map, are also Gscodes for some s.

SelfDual Codes using Bisymmetric Matrices and Group RingsIn this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F4 + uF4 together with the well known extension and neighbour methods, we construct new selfdual codes of length 68: In particular, we find 41 new codes of length 68 that were not known in the literature before.

2^n Bordered Constructions of SelfDual codes from Group RingsSelfdual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary selfdual codes. In this paper, we introduce a new bordered construction over group rings for selfdual codes by combining many of the previously used techniques. The purpose of this is to construct selfdual codes that were missed using classical construction techniques by constructing selfdual codes with diﬀerent automorphism groups. We apply the technique to codes over ﬁnite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary selfdual codes. In particular, we construct some extremal selfdual codes length 64 and 68, constructing 30 new extremal selfdual codes of length 68.

An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated spacetime white noiseWe consider the strong convergence of the numerical methods for solving stochastic subdiffusion problem driven by an integrated spacetime white noise. The time fractional derivative is approximated by using the L1 scheme and the time fractional integral is approximated with the Lubich's first order convolution quadrature formula. We use the Euler method to approximate the noise in time and use the truncated series to approximate the noise in space. The spatial variable is discretized by using the linear finite element method. Applying the idea in Gunzburger \et (Math. Comp. 88(2019), pp. 17151741), we express the approximate solutions of the fully discrete scheme by the convolution of the piecewise constant function and the inverse Laplace transform of the resolvent related function. Based on such convolution expressions of the approximate solutions, we obtain the optimal convergence orders of the fully discrete scheme in spatial multidimensional cases by using the Laplace transform method and the corresponding resolvent estimates.

New binary selfdual codes via a generalization of the four circulant constructionIn this work, we generalize the four circulant construction for selfdual codes. By applying the constructions over the alphabets $\mathbb{F}_2$, $\mathbb{F}_2+u\mathbb{F}_2$, $\mathbb{F}_4+u\mathbb{F}_4$, we were able to obtain extremal binary selfdual codes of lengths 40, 64 including new extremal binary selfdual codes of length 68. More precisely, 43 new extremal binary selfdual codes of length 68, with rare new parameters have been constructed.

Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth DataTwo higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Gr\"unwaldLetnikov formulae introduced in Tian et al. [Math. Comp. 84 (2015), pp. 27032727]. After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders $O(k^2)$ and $ O(k^3)$, respectively for any fixed time $t$ for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.

A Modified Bordered Construction for SelfDual Codes from Group RingsWe describe a bordered construction for selfdual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary selfdual codes of various lengths. In particular we find a new extremal binary selfdual code of length 78.

Finitetime blowup of a nonlocal stochastic parabolic problemThe main aim of the current work is the study of the conditions under which (finitetime) blowup of a nonlocal stochastic parabolic problem occurs. We first establish the existence and uniqueness of the localintime weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noiseinduced blowup. In the second part we first prove the $C^{1}$spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (nonlocal) drift term induced blowup result. In the last part of the paper, we present a method which provides an upper bound of the probability of (nonlocal) drift term induced blowup.

Gcodes over Formal Power Series Rings and Finite Chain RingsIn this work, we define $G$codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$code is again a $G$code in this setting. We study the projections and lifts of $G$codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$adic codes over $R_\infty$ to $\gamma$adic $G$codes over the same ring. We also study $G$codes over principal ideal rings.

New Extremal SelfDual Binary Codes of Length 68 via Composite Construction, F2 + uF2 Lifts, Extensions and NeighborsWe describe a composite construction from group rings where the groups have orders 16 and 8. This construction is then applied to find the extremal binary selfdual codes with parameters [32, 16, 8] or [32, 16, 6]. We also extend this composite construction by expanding the search field which enables us to find more extremal binary selfdual codes with the above parameters and with different orders of automorphism groups. These codes are then lifted to F2 + uF2, to obtain extremal binary images of codes of length 64. Finally, we use the extension method and neighbor construction to obtain new extremal binary selfdual codes of length 68. As a result, we obtain 28 new codes of length 68 which were not known in the literature before.

Modified Quadratic Residue Constructions and New Exermal Binary SelfDual Codes of Lengths 64, 66 and 68In this work we consider modiﬁed versions of quadratic double circulant and quadratic bordered double circulant constructions over the binary ﬁeld and the rings F2 +uF2 and F4 +uF4 for diﬀerent prime values of p. Using these constructions with extensions and neighbors we are able to construct a number of extremal binary selfdual codes of diﬀerent lengths with new parameters in their weight enumerators. In particular we construct 2 new codes of length 64, 4 new codes of length 66 and 14 new codes of length 68. The binary generator matrices of the new codes are available online at [8].

High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problemWe use the generalized L1 approximation for the Caputo fractional derivative, the secondorder fractional quadrature rule approximation for the integral term, and a classical CrankNicolson alternating direction implicit (ADI)scheme for the time discretization of a new twodimensional (2D) fractionalintegrodifferential equation, in combination with a space discretization by anarbitraryorder orthogonal spline collocation (OSC) method. The stability of aCrankNicolson ADI OSC scheme is rigourously established, and error estimateis also derived. Finally, some numerical tests are given