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It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the structure of dually qOACs and compute their weight distributions, generalized weights, and associated q-polymatroids.

In this paper, several classes of three-weight ternary linear codes from non-weakly regular dual-bent functions are constructed based on a generic construction method. Instead of the whole space, we use the subspaces $B_+(f)$ or $B_-(f)$ associated with a ternary non-weakly regular dual-bent function $f$. Unusually, we use the pre-image sets of the dual function $f^*$ in $B_+(f)$ or $B_-(f)$ as the defining sets of the corresponding codes. Since the size of the defining sets of the constructed codes is flexible, it enables us to construct several codes with different parameters for a fixed dimension. We represent the weight distribution of the constructed codes. We also give several examples.

A code $\mathcal{C} \subseteq \{0, 1, 2\}^n$ is said to be trifferent with length $n$ when for any three distinct elements of $\mathcal{C}$ there exists a coordinate in which they all differ. Defining $\mathcal{T}(n)$ as the maximum cardinality of trifferent codes with length $n$, $\mathcal{T}(n)$ is unknown for $n \ge 5$. In this note, we use an optimized search algorithm to show that $\mathcal{T}(5) = 10$ and $\mathcal{T}(6) = 13$.

We show that solution to the Hermite-Pad\'{e} type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Pad\'{e} approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Pad\'{e} problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. In turn, these lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality.

The numerical analysis of causal fermion systems is advanced by employing differentiable programming methods. The causal action principle for weighted counting measures is introduced for general values of the integer parameters $f$ (the particle number), $n$ (the spin dimension) and $m$ (the number of spacetime points). In the case $n=1$, the causal relations are clarified geometrically in terms of causal cones. Discrete Dirac spheres are introduced as candidates for minimizers for large $m$ in the cases $n=1, f=2$ and $n=2, f=4$. We provide a thorough numerical analysis of the causal action principle for weighted counting measures for large $m$ in the cases $n=1,2$ and $f=2,3,4$. Our numerical findings corroborate that all minimizers for large $m$ are good approximations of the discrete Dirac spheres. In the example $n=1, f=3$ it is explained how numerical minimizers can be visualized by projected spacetime plots. Methods and prospects are discussed to numerically investigate settings in which hitherto no analytic candidates for minimizers are known.

The aim of this note is to state a couple of general results about the properties of the penalized maximum likelihood estimators (pMLE) and of the posterior distribution for parametric models in a non-asymptotic setup and for possibly large or even infinite parameter dimension. We consider a special class of stochastically linear smooth (SLS) models satisfying two major conditions: the stochastic component of the log-likelihood is linear in the model parameter, while the expected log-likelihood is a smooth function. The main results simplify a lot if the expected log-likelihood is concave. For the pMLE, we establish a number of finite sample bounds about its concentration and large deviations as well as the Fisher and Wilks expansion. The later results extend the classical asymptotic Fisher and Wilks Theorems about the MLE to the non-asymptotic setup with large parameter dimension which can depend on the sample size. For the posterior distribution, our main result states a Gaussian approximation of the posterior which can be viewed as a finite sample analog of the prominent Bernstein--von Mises Theorem. In all bounds, the remainder is given explicitly and can be evaluated in terms of the effective sample size and effective parameter dimension. The results are dimension and coordinate free. In spite of generality, all the presented bounds are nearly sharp and the classical asymptotic results can be obtained as simple corollaries. An interesting case of logit regression with smooth or truncation priors is used to specify the results and to explain the main notions.

We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of "local" problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete Parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the Parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our Time Parallel Algorithm (TPA) and Space-Time Parallel Algorithm (STPA) respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.

A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and then to appeal to functoriality. However, we often lack such maps in real data; instead, we must rely on a cross-dissimilarity measure between our observations of a system and a reference. In this paper, we develop a pair of computational homological algebra approaches for relating persistent homology classes and barcodes: persistent extension, which enumerates potential relations between cycles from two complexes built on the same vertex set, and the method of analogous bars, which utilizes persistent extension and the witness complex built from a cross-dissimilarity measure to provide relations across systems. We provide an implementation of these methods and demonstrate their use in comparing cycles between two samples from the same metric space and determining whether topology is maintained or destroyed under clustering and dimensionality reduction.

This paper uses the concept of algorithmic efficiency to present a unified theory of intelligence. Intelligence is defined informally, formally, and computationally. I introduce the concept of Dimensional complexity in algorithmic efficiency and deduce that an optimally efficient algorithm has zero Time complexity, zero Space complexity, and an infinite Dimensional complexity. This algorithm is then used to generate the number line.