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Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual $\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the largest possible minimum Euclidean weight for a given length. We introduce a doubling method for constructing a Type II $\mathbb{Z}_{2k}$-code of length $n$ from a known Type II $\mathbb{Z}_{2k}$-code of length $n$. Based on this method, we develop an algorithm to construct new extremal Type II $\mathbb{Z}_8$-codes starting from an extremal Type II $\mathbb{Z}_8$-code of type $(\frac{n}{2},0,0)$ with an extremal $\mathbb{Z}_4$-residue code and length $24, 32$ or $40$. We construct at least ten new extremal Type II $\mathbb{Z}_8$-codes of length $32$ and type $(15,1,1)$. Extremal Type II $\mathbb{Z}_8$-codes of length $32$ of this type were not known before. Moreover, the binary residue codes of the constructed extremal $\mathbb{Z}_8$-codes are optimal $[32,15]$ binary codes.

When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time $t^{n+\beta}$ with $\beta > 1$ being a tunable parameter. These new schemes, with a suitable $\beta$, allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes, and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.

In this paper, for any fixed positive integers $t$ and $q>2$, we construct $q$-ary codes correcting a burst of at most $t$ deletions with redundancy $\log n+8\log\log n+o(\log\log n)+\gamma_{q,t}$ bits and near-linear encoding/decoding complexity, where $n$ is the message length and $\gamma_{q,t}$ is a constant that only depends on $q$ and $t$. In previous works there are constructions of such codes with redundancy $\log n+O(\log q\log\log n)$ bits or $\log n+O(t^2\log\log n)+O(t\log q)$. The redundancy of our new construction is independent of $q$ and $t$ in the second term.

We numerically study the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions modeled by Bosonic Embedded Gaussian Orthogonal [BEGOE($k$)] and Unitary [BEGUE($k$)] random matrix Ensembles. Following the recently established result that the $q$-normal describes the smooth form of the eigenvalue density of the $k$-body embedded ensembles, the first four moments of the distribution of lowest eigenvalues have been analyzed as a function of the $q$ parameter, with $q \sim 1$ for $k = 1$ and $q = 0$ for $k = m$; $m$ being the number of bosons. Our results show the distribution exhibits a smooth transition from Gaussian like for $q$ close to 1 to a modified Gumbel like for intermediate values of $q$ to the well-known Tracy-Widom distribution for $q=0$.

This paper studies the $p$-biharmonic equation on graphs, which arises in point cloud processing and can be interpreted as a natural extension of the graph $p$-Laplacian from the perspective of hypergraph. The asymptotic behavior of the solution is investigated when the random geometric graph is considered and the number of data points goes to infinity. We show that the continuum limit is an appropriately weighted $p$-biharmonic equation with homogeneous Neumann boundary conditions. The result relies on the uniform $L^p$ estimates for solutions and gradients of nonlocal and graph Poisson equations. The $L^\infty$ estimates of solutions are also obtained as a byproduct.

We suggest new closely related methods for numerical inversion of $Z$-transform and Wiener-Hopf factorization of functions on the unit circle, based on sinh-deformations of the contours of integration, corresponding changes of variables and the simplified trapezoid rule. As applications, we consider evaluation of high moments of probability distributions and construction of causal filters. Programs in Matlab running on a Mac with moderate characteristics achieves the precision E-14 in several dozen of microseconds and E-11 in several milliseconds, respectively.

We study the communication complexity of $(\Delta + 1)$ vertex coloring, where the edges of an $n$-vertex graph of maximum degree $\Delta$ are partitioned between two players. We provide a randomized protocol which uses $O(n)$ bits of communication and ends with both players knowing the coloring. Combining this with a folklore $\Omega(n)$ lower bound, this settles the randomized communication complexity of $(\Delta + 1)$-coloring up to constant factors.

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph $D$ is $k$-dicritical if $\vec{\chi}(D) = k$ and each proper subdigraph $H$ of $D$ satisfies $\vec{\chi}(H) < k$. For integers $k$ and $n$, we define $d_k(n)$ (respectively $o_k(n)$) as the minimum number of arcs possible in a $k$-dicritical digraph (respectively oriented graph). Kostochka and Stiebitz have shown that $d_4(n) \geq \frac{10}{3}n -\frac{4}{3}$. They also conjectured that there is a constant $c$ such that $o_k(n) \geq cd_k(n)$ for $k\geq 3$ and $n$ large enough. This conjecture is known to be true for $k=3$ (Aboulker et al.). In this work, we prove that every $4$-dicritical oriented graph on $n$ vertices has at least $(\frac{10}{3}+\frac{1}{51})n-1$ arcs, showing the conjecture for $k=4$. We also characterise exactly the $k$-dicritical digraphs on $n$ vertices with exactly $\frac{10}{3}n -\frac{4}{3}$ arcs.

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least integer $k$ for which $D$ has a coloring with $k$ colors such that there is no monochromatic directed cycle in $D$. The digraphs considered here are finite and may have antiparallel arcs, but no parallel arcs. A digraph $D$ is called $k$-critical if each proper subdigraph $D'$ of $D$ satisfies $\vec{\chi}(D')<\vec{\chi}(D)=k$. For integers $k$ and $n$, let $\overrightarrow{\mathrm{ext}}(k,n)$ denote the minimum number of arcs possible in a $k$-critical digraph of order $n$. It is easy to show that $\overrightarrow{\mathrm{ext}}(2,n)=n$ for all $n\geq 2$, and $\overrightarrow{\mathrm{ext}}(3,n)\geq 2n$ for all possible $n$, where equality holds if and only if $n$ is odd and $n\geq 3$. As a main result we prove that if $n, k$ and $p$ are integers with $n=k+p$ and $2\leq p \leq k-1$, then $\overrightarrow{\mathrm{ext}}(k,n)=2({\binom{n}{2}} - (p^2+1))$, and we give an exact characterisation of $k$-critical digraphs for which equality holds. This generalizes a result about critical graphs obtained in 1963 by Tibor Gallai.

We study the problem of testing $C_k$-freeness ($k$-cycle-freeness) for fixed constant $k > 3$ in graphs with bounded arboricity (but unbounded degrees). In particular, we are interested in one-sided error algorithms, so that they must detect a copy of $C_k$ with high constant probability when the graph is $\epsilon$-far from $C_k$-free. We next state our results for constant arboricity and constant $\epsilon$ with a focus on the dependence on the number of graph vertices, $n$. The query complexity of all our algorithms grows polynomially with $1/\epsilon$. (1) As opposed to the case of $k=3$, where the complexity of testing $C_3$-freeness grows with the arboricity of the graph but not with the size of the graph (Levi, ICALP 2021) this is no longer the case already for $k=4$. We show that $\Omega(n^{1/4})$ queries are necessary for testing $C_4$-freeness, and that $\widetilde{O}(n^{1/4})$ are sufficient. The same bounds hold for $C_5$. (2) For every fixed $k \geq 6$, any one-sided error algorithm for testing $C_k$-freeness must perform $\Omega(n^{1/3})$ queries. (3) For $k=6$ we give a testing algorithm whose query complexity is $\widetilde{O}(n^{1/2})$. (4) For any fixed $k$, the query complexity of testing $C_k$-freeness is upper bounded by ${O}(n^{1-1/\lfloor k/2\rfloor})$. Our $\Omega(n^{1/4})$ lower bound for testing $C_4$-freeness in constant arboricity graphs provides a negative answer to an open problem posed by (Goldreich, 2021).