Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $\rho^l$ and the scalar multiplications of $\mathcal{C}$, where $\rho$ denotes the standard cyclic shift. In this paper, we find an explicit formula of orbits of $G$ on $\mathcal{C}\setminus \{\mathbf{0}\}$. Consequently, an explicit upper bound on the number of nonzero weights of $\mathcal{C}$ is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. If $\mathcal{C}$ is a one-generator quasi-cyclic code, a tighter upper bound on the number of nonzero weights of $\mathcal{C}$ is obtained by considering a larger automorphism subgroup which is generated by the multiplier, $\rho^l$ and the scalar multiplications of $\mathcal{C}$. In particular, we list some examples to show the bounds are tight. Our main result improves and generalizes some of the results in \cite{M2}.
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In this paper, we propose a new method for constructing $1$-perfect mixed codes in the Cartesian product $\mathbb{F}_{n} \times \mathbb{F}_{q}^n$, where $\mathbb{F}_{n}$ and $\mathbb{F}_{q}$ are finite fields of orders $n = q^m$ and $q$. We consider generalized Reed-Muller codes of length $n = q^m$ and order $(q - 1)m - 2$. Codes whose parameters are the same as the parameters of generalized Reed-Muller codes are called Reed-Muller-like codes. The construction we propose is based on partitions of distance-2 MDS codes into Reed-Muller-like codes of order $(q - 1)m - 2$. We construct a set of $q^{q^{cn}}$ nonequivalent 1-perfect mixed codes in the Cartesian product $\mathbb{F}_{n} \times \mathbb{F}_{q}^{n}$, where the constant $c$ satisfies $c < 1$, $n = q^m$ and $m$ is a sufficiently large positive integer. We also prove that each $1$-perfect mixed code in the Cartesian product $\mathbb{F}_{n} \times \mathbb{F}_{q}^n$ corresponds to a certain partition of a distance-2 MDS code into Reed-Muller-like codes of order $(q - 1)m - 2$.
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.
We consider polynomials of the form $\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\lambda$ is an integer partition, $\operatorname{s}_\lambda$ is the Schur polynomial associated to $\lambda$, and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. We represent $\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$ as a quotient whose the denominator is the determinant of the confluent Vandermonde matrix, and the numerator is the determinant of some generalized confluent Vandermonde matrix. We give three algebraic proofs of this formula.
An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semi-discretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.
Given a matrix-valued function $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, with complex matrices $A_i$ and $f_i(\lambda)$ entire functions for $i=1,\ldots,d$, we discuss a method for the numerical approximation of the distance to singularity of $\mathcal{F}(\lambda)$. The closest singular matrix-valued function $\widetilde{\mathcal{F}}(\lambda)$ with respect to the Frobenius norm is approximated using an iterative method. The property of singularity on the matrix-valued function is translated into a numerical constraint for a suitable minimization problem. Unlike the case of matrix polynomials, in the general setting of matrix-valued functions the main issue is that the function $\det ( \widetilde{\mathcal{F}}(\lambda) )$ may have an infinite number of roots. An important feature of the numerical method consists in the possibility of addressing different structures, such as sparsity patterns induced by the matrix coefficients, in which case the search of the closest singular function is restricted to the class of functions preserving the structure of the matrices.
Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when $q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the $e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when $q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.
We investigate completions of partial combinatory algebras (pcas), in particular of Kleene's second model $\mathcal{K}_2$ and generalizations thereof. We consider weak and strong notions of embeddability and completion that have been studied before. By a result of Klop it is known that not every pca has a strong completion. The study of completions of $\mathcal{K}_2$ has as corollaries that weak and strong embeddings are different, and that every countable pca has a weak completion. We then consider generalizations of $\mathcal{K}_2$ for larger cardinals, and use these to show that it is consistent that every pca has a weak completion.
A {\em packing coloring} of a graph $G$ is a mapping assigning a positive integer (a color) to every vertex of $G$ such that every two vertices of color $k$ are at distance at least $k+1$. The least number of colors needed for a packing coloring of $G$ is called the {\em packing chromatic number} of $G$. In this paper, we continue the study of the packing chromatic number of hypercubes and we improve the upper bounds reported by Torres and Valencia-Pabon ({\em P. Torres, M. Valencia-Pabon, The packing chromatic number of hypercubes, Discrete Appl. Math. 190--191 (2015), 127--140}) by presenting recursive constructions of subsets of distant vertices making use of the properties of the extended Hamming codes. We also answer in negative a question on packing coloring of Cartesian products raised by Bre\v{s}ar, Klav\v{z}ar, and Rall ({\em Problem 5, Bre\v{s}ar et al., On the packing chromatic number of Cartesian products, hexagonal lattice, and trees. Discrete Appl. Math. 155 (2007), 2303--2311.}).
Let $L$ be a set of $n$ axis-parallel lines in $\mathbb{R}^3$. We are are interested in partitions of $\mathbb{R}^3$ by a set $H$ of three planes such that each open cell in the arrangement $\mathcal{A}(H)$ is intersected by as few lines from $L$ as possible. We study such partitions in three settings, depending on the type of splitting planes that we allow. We obtain the following results. $\bullet$ There are sets $L$ of $n$ axis-parallel lines such that, for any set $H$ of three splitting planes, there is an open cell in $\mathcal{A}(H)$ that intersects at least~$\lfloor n/3 \rfloor-1 \approx \frac{1}{3}n$ lines. $\bullet$ If we require the splitting planes to be axis-parallel, then there are sets $L$ of $n$ axis-parallel lines such that, for any set $H$ of three splitting planes, there is an open cell in $\mathcal{A}(H)$ that intersects at least $\frac{3}{2}\lfloor n/4 \rfloor -1 \approx \left( \frac{1}{3}+\frac{1}{24}\right) n$ lines. Furthermore, for any set $L$ of $n$ axis-parallel lines, there exists a set $H$ of three axis-parallel splitting planes such that each open cell in $\mathcal{A}(H)$ intersects at most $\frac{7}{18} n = \left( \frac{1}{3}+\frac{1}{18}\right) n$ lines. $\bullet$ For any set $L$ of $n$ axis-parallel lines, there exists a set $H$ of three axis-parallel and mutually orthogonal splitting planes, such that each open cell in $\mathcal{A}(H)$ intersects at most $\lceil \frac{5}{12} n \rceil \approx \left( \frac{1}{3}+\frac{1}{12}\right) n$ lines.
Given a Binary Decision Diagram $B$ of a Boolean function $\varphi$ in $n$ variables, it is well known that all $\varphi$-models can be enumerated in output polynomial time, and in a compressed way (using don't-care symbols). We show that all $N$ many $\varphi$-models of fixed Hamming-weight $k$ can be enumerated as well in time polynomial in $n$ and $|B|$ and $N$. Furthermore, using novel wildcards, again enables a compressed enumeration of these models.
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