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We prove the switching equivalence of the strongly regular polar graphs $NO^\pm(4m,2)$, $NO^\mp(2m+1,4)$, and $\Gamma(O^\mp(4m,2))$ plus an isolated vertex by giving an analytic description for them and their associated two-graphs.

Let $C$ be a linear code of length $n$ and dimension $k$ over the finite field $\mathbb{F}_{q^m}$. The trace code $\mathrm{Tr}(C)$ is a linear code of the same length $n$ over the subfield $\mathbb{F}_q$. The obvious upper bound for the dimension of the trace code over $\mathbb{F}_q$ is $mk$. If equality holds, then we say that $C$ has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let $C_{\mathbf{a}}$ denote the code obtained from $C$ and a multiplier vector $\mathbf{a}\in (\mathbb{F}_{q^m})^n$. In this paper, we give a lower bound for the probability that a random multiplier vector produces a code $C_{\mathbf{a}}$ of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever $n\geq m(k+h)$, where $h\geq 0$ is the Singleton defect of $C$. For the extremal case $n=m(h+k)$, numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.

In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with $k$ close to the order of the plane, is considered a hard problem. A natural candidate to be a small $(k,n)$-arc with few characters is the set $\Omega(\mathcal{C})$ of the points of a plane curve $\mathcal{C}$ of degree $n$ (containing no linear components) such that some line meets $\mathcal{C}$ transversally in the plane, i.e. in $n$ pairwise distinct points. Let $\mathcal{C}$ be either the Hermitian curve of degree $q+1$ in $\mathrm{PG}(2,q^{2r})$ with $r\ge 1$, or the rational BKS curve of degree $q+1$ in $\mathrm{PG}(2,q^r)$ with $q$ odd and $r\ge 1$. Then $\Omega(\mathcal{C})$ has four and seven characters, respectively. Furthermore, $\Omega(\mathcal{C})$ is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that $\Omega(\mathcal{C})$ is complete for $r\ge 4$. For the rational BKS curve, $\Omega(\mathcal{C})$ is complete if and only if $r$ is even. If $r$ is odd then the uncovered points by the $(q+1)$-secants to $\Omega(\mathcal{C})$ are exactly the points in $\mathrm{PG}(2,q)$ not lying in $\Omega(\mathcal{C})$. Adding those points to $\Omega(\mathcal{C})$ produces a complete $(k,q+1)$-arc in $\mathrm{PG}(2,q^r)$, with $k=q^r+q$. The above results do not hold true for $r=2$ and there remain open the case $r=3$ for the Hermitian curve, and the cases $r=3,4$ for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for $\mathrm{PGL}(2,q)$.

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra $\mathfrak{g}$ with $\operatorname{dim}[\mathfrak{g},\mathfrak{g}]=1$ is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra $\mathfrak{l}_{2n+1}^{J_1}$, where $J_1$ is the $n \times n$ Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks., Comment: Final version, accepted for publication

The vectorial nonlinearity of a vector valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet proved a lower bound in terms of the differential uniformity. In this paper, we improve Carlet's lower bound. Our method is elementary, it relies on the fact that the level sets of an APN functions are Sidon sets. We give a survey on Sidon sets in elementary abelian 2-groups. We study the completeness problem of Sidon sets obtained from hyperbolas and ellipses of the finite affine plane.

It is known that correlation-immune (CI) Boolean functions used in the framework of side-channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple binary orthogonal arrays of strengths 2 and 3; we show that this minimum is the same in such case as for all OA, and we extend this observation to some OA of strengths $4$ and $5$. This allows us to reply positively, in the case of strengths 2 and 3, to a question raised by the first author and X. Chen on the monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to partially reply positively to the same question in the case of strengths 4 and 5., Comment: Des. Codes Cryptogr. (2022)

We show that no orthogonal arrays $OA(16 \lambda, 11, 2,4)$ exist with $\lambda=6$ and $\lambda=7$. This solves an open problem of the NSUCRYPTO Olympiad 2018. Our result allows us to determine the minimum weights of certain higher-order correlation-immune Boolean functions.

We show that the Ree unital $\mathcal{R}(q)$ has an embedding in a projective plane over a field $F$ if and only if $q=3$ and $\mathbb{F}_8$ is a subfield of $F$. In this case, the embedding is unique up to projective linear transformations. Besides elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.

In this paper, we study the behavior of the true dimension of the subfield subcodes of Hermitian codes. Our motivation is to use these classes of linear codes to improve the parameters of the McEliece cryptosystem, such that key size and security level. The McEliece scheme is one of the promising alternative cryptographic schemes to the current public key schemes since in the last four decades, they resisted all known quantum computing attacks. By analyzing computational data series of true dimension, we concluded that they can be estimated by the extreme value distribution function.

Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals., Comment: Revised version based on remarks of anonymous referee

We show that $\Gamma$-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences. Further, we recover some of Hetyei's results on alternating acyclic tournaments. Finally, we present an interesting result in the case of certain other special shapes., Comment: 17 pages, 8 figures

A group $G$ has a Frobenius graphical representation (GFR) if there is a simple graph $\varGamma$ whose full automorphism group is isomorphic to $G$ and it acts on vertices as a Frobenius group. In particular, any group $G$ with GFR is a Frobenius group and $\varGamma$ is a Cayley graph. The existence of an infinite family of groups with GFR whose Frobenius kernel is a non-abelian $2$-group has been an open question. In this paper, we give a positive answer by showing that the Higman group $A(f,q_0)$ has a GFR for an infinite sequence of $f$ and $q_0$.

Subfield subcodes of algebraic-geometric codes are good candidates for the use in post-quantum cryptosystems, provided their true parameters such as dimension and minimum distance can be determined. In this paper we present new values of the true dimension of subfield subcodes of $1$--point Hermitian codes, including the case when the subfield is not binary.

The concept of full points of abstract unitals has been introduced by Korchm\'aros, Siciliano and Sz\H{o}nyi as a tool for the study of projective embeddings of abstract unitals. In this paper we give a more detailed description of the combinatorial and geometric structure of the sets of full points in abstract unitals of finite order.

The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures.

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of ${\rm PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre-Korchm\'aros on Desargues configurations with perspective triangles inscribed in a conic.

We present a new method for the study of hemisystems of the Hermitian surface $\mathcal{U}_3$ of $PG(3,q^2)$. The basic idea is to represent generator-sets of $\mathcal{U}_3$ by means of a maximal curve naturally embedded in $\mathcal{U}_3$ so that a sufficient condition for the existence of hemisystems may follow from results about maximal curves and their automorphism groups. In this paper we obtain a hemisystem in $\ PG(3,p^2)$ for each $p$ prime of the form $p=1+16n^2$ with an integer $n$. Since the famous Landau's conjecture dating back to 1904 is still to be proved (or disproved), it is unknown whether there exists an infinite sequence of such primes. What is known so far is that just $18$ primes up to $51000$ with this property exist, namely $17,257,401,577, 1297,1601, 3137, 7057,13457,14401,15377,24337,25601,30977,$ $ 32401,33857,41617,50177.$ The scarcity of such primes seems to confirm that hemisystems of $\mathcal{U}_3$ are rare objects.

In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the characteristic $\mathbb{K}$, we classify group-labeled light dual multinets with lines of length least $9$.

We construct a new bijection between the set of $n\times k$ $0$-$1$ matrices with no three $1$'s forming a $\Gamma$ configuration and the set of $(n,k)$-Callan sequences, a simple structure counted by poly-Bernoulli numbers. We give two applications of this result: We derive the generating function of $\Gamma$-free matrices, and we give a new bijective proof for an elegant result of Aval et al. that states that the number of complete non-ambiguous forests with $n$ leaves is equal to the number of pairs of permutations of $\{1,\dots,n\}$ with no common rise.

Right Bol loops are loops satisfying the identity $((zx)y)x = z((xy)x)$, and right Bruck loops are right Bol loops satisfying the identity $(xy)^{-1} = x^{-1}y^{-1}$. Let $p$ and $q$ be odd primes such that $p>q$. Advancing the research program of Niederreiter and Robinson from $1981$, we classify right Bol loops of order $pq$. When $q$ does not divide $p^2-1$, the only right Bol loop of order $pq$ is the cyclic group of order $pq$. When $q$ divides $p^2-1$, there are precisely $(p-q+4)/2$ right Bol loops of order $pq$ up to isomorphism, including a unique nonassociative right Bruck loop $B_{p,q}$ of order $pq$. Let $Q$ be a nonassociative right Bol loop of order $pq$. We prove that the right nucleus of $Q$ is trivial, the left nucleus of $Q$ is normal and is equal to the unique subloop of order $p$ in $Q$, and the right multiplication group of $Q$ has order $p^2q$ or $p^3q$. When $Q=B_{p,q}$, the right multiplication group of $Q$ is isomorphic to the semidirect product of $\mathbb{Z}_p\times \mathbb{Z}_p$ with $\mathbb{Z}_q$. Finally, we offer computational results as to the number of right Bol loops of order $pq$ up to isotopy., Comment: 23 pages; to appear in J. Algebra

In this paper we propose a framework for automated forecasting of energy-related time series using open access data from European Network of Transmission System Operators for Electricity (ENTSO-E). The framework provides forecasts for various European countries using publicly available historical data only. Our solution was benchmarked using the actual load data and the country provided estimates (where available). We conclude that the proposed system can produce timely forecasts with comparable prediction accuracy in a number of cases. We also investigate the probabilistic case of forecasting - that is, providing a probability distribution rather than a simple point forecast - and incorporate it into a web based API that provides quick and easy access to reliable forecasts., Comment: 6 pages, 6 figures, IEEE Energycon 2016, Leuven, Belgium

A \textit{$3$-net} of order $n$ is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around $3$-nets (embedded) in a projective plane $PG(2,K)$, defined over a field $K$ of characteristic $p$, arose from algebraic geometry. It is not difficult to find $3$-nets in $PG(2,K)$ as far as $0n$. Under this condition, the known families are characterized as the only $3$-nets in $PG(2,K)$ which can be coordinatized by a group. In this paper we deal with $3$-nets in $PG(2,K)$ which can be coordinatized by a diassociative loop $G$ but not by a group. We prove two structural theorems on $G$. As a corollary, if $G$ is commutative then every non-trivial element of $G$ has the same order, and $G$ has exponent $2$ or $3$. We also discuss the existence problem for such $3$-nets.

Gy\'arf\'as investigated a geometric Ramsey problem on convex, separated, balanced, geometric $K_{n,n}$. This led to appealing extremal problem on square $0$-$1$ matrices. Gy\'arf\'as conjectured that any $0$-$1$ matrix of size $n\times n$ has a staircase of size $n-1$. We introduce the non-symmetric version of Gy\'arf\'as' problem. We give upper bounds and in certain range matching lower bound on the corresponding extremal function. In the square/balanced case we improve the $(4/5+\epsilon)n$ lower bound of Cai, Gy\'arf\'as et al. to $5n/6-7/12$. We settle the problem when instead of considering maximum staircases we deal with the sum of the size of the longest $0$- and $1$-staircases., Comment: 10 pages

We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most two. $6$-divisible Moufang loops with commuting inner mappings have nilpotency class at most two. There is a Moufang loop of order $2^{14}$ with commuting inner mappings and of nilpotency class three.

We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the $15$ groups of order $81$, there are $5$ nonassociative Moufang loops of order $81$, $2$ of which are commutative.

This is a companion to our lectures GAP and loops, to be delivered at the Workshops Loops 2007, Prague, Czech Republic. In the lectures we introduce the GAP package LOOPS, describe its capabilities, and explain in detail how to use it. In this paper we first outline the philosophy behind the package and its main features, and then we focus on three particular computational problems: construction of loop isomorphisms, classification of small Frattini Moufang loops of order 64, and the search for loops of nilpotency class higher than two with an abelian inner mapping group. In particular, this is not a user's manual for LOOPS, which can be downloaded from the distribution website of LOOPS.

In this paper we implement a Local Linear Regression Ensemble Committee (LOLREC) to predict 1-day-ahead returns of 453 assets form the S&P500. The estimates and the historical returns of the committees are used to compute the weights of the portfolio from the 453 stock. The proposed method outperforms benchmark portfolio selection strategies that optimize the growth rate of the capital. We investigate the effect of algorithm parameter m: the number of selected stocks on achieved average annual yields. Results suggest the algorithm's practical usefulness in everyday trading., Comment: RCOMM 2012: Rapidminer Community Meeting and Conference

Energy price forecasting is a relevant yet hard task in the field of multi-step time series forecasting. In this paper we compare a well-known and established method, ARMA with exogenous variables with a relatively new technique Gradient Boosting Regression. The method was tested on data from Global Energy Forecasting Competition 2014 with a year long rolling window forecast. The results from the experiment reveal that a multi-model approach is significantly better performing in terms of error metrics. Gradient Boosting can deal with seasonality and auto-correlation out-of-the box and achieve lower rate of normalized mean absolute error on real-world data., Comment: 10 pages, 5 figures, KES-IDT 2015 conference. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-19857-6_7

The intrinsic structure of binary fields poses a challenging complexity problem from both hardware and software point of view. Motivated by applications to modern cryptography, we describe some simple techniques aimed at performing computations over binary fields using systems with limited resources. This is particularly important when such computations must be carried out by means of very small and simple machines. The algorithms described in the present paper provide an increased efficiency in computations, when compared to the previously known algorithms for the arithmetic over prime fields.

In this short paper, we survey the results on commutative automorphic loops and give a new construction method. Using this method, we present new classes of commutative automorphic loops of exponent 2 with trivial center.

For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of finite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the isomorphism of the corresponding translation planes.

Matthews and Michel investigated the minimum distances in certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the Hermitian function field. We determine the Weierstrass gap sequence $G(P)$ where $P$ is a degree 3 place, and compute the Matthews and Michel bound with the corresponding improvement. We show more improvements using a different approach based on geometry. We also compare our results with the true values of the minimum distances of Hermitian 1-point codes, as well as with estimates due Xing and Chen.

Shapiro proved an elegant convolution formula involving Catalan numbers of even index. This paper gives a simple combinatorial proof of his formula. In addition, we show that it is equivalent with the alternating convolution formula of central binomial coefficients.

We prove that every finite, commutative automorphic loop is solvable. We also prove that every finite, automorphic 2-loop is solvable. The main idea of the proof is to associate a simple Lie algebra of characteristic 2 to a hypothetical finite simple commutative automorphic loop. The "crust of a thin sandwich" theorem of Zel'manov and Kostrikin leads to a contradiction., Comment: v.2: minor revisions suggested by referee

In this paper, we investigate dual 3-nets realizing the groups $C_3 \times C_3$, $C_2 \times C_4$, $\Alt_4$ and that can be embedded in a projective plane $PG(2,\mathbb K)$, where $\mathbb K$ is an algebraically closed field. We give a symbolically verifiable computational proof that every dual 3-net realizing the groups $C_3 \times C_3$ and $C_2 \times C_4$ is algebraic, namely, that its points lie on a plane cubic. Moreover, we present two computer programs whose calculations show that the group $\Alt_4$ cannot be realized if the characteristic of $\mathbb K$ is zero., Comment: 23 pages, Maple and Singular program codes in Appendices