On the number of n-ary quasigroups of finite order

Let $Q(n,k)$ be the number of $n$-ary quasigroups of order $k$. We derive a recurrent formula for Q(n,4). We prove that for all $n\geq 2$ and $k\geq 5$ the following inequalities hold: $({k-3}/2)^{n/2}(\frac{k-1}2)^{n/2} < log_2 Q(n,k) \leq {c_k(k-2)^{n}}$, where $c_k$ does not depend on $n$. So, the upper asymptotic bound for $Q(n,k)$ is improved for any $k\geq 5$ and the lower bound is improved for odd $k\geq 7$. Keywords: n-ary quasigroup, latin cube, loop, asymptotic estimate, component, latin trade.Comment: english 9pp, russian 9pp. v.2: corrected: initial data for recursion; added: Appendix with progra

Propelinear 1-perfect codes from quadratic functions

Perfect codes obtained by the Vasil'ev--Sch\"onheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least $\exp(cN^2)$ propelinear $1$-perfect codes of length $N$ over an arbitrary finite field, while an upper bound on the number of transitive codes is $\exp(C(N\ln N)^2)$. Keywords: perfect code, propelinear code, transitive code, automorphism group, Boolean function.Comment: 4 IEEE pages. v2: minor revision, + upper bound (Sect. III.B), +remarks (Sect. V.A); v3: minor revision, + length 15 (Sect. V.B

n-Ary quasigroups of order 4

We characterize the set of all N-ary quasigroups of order 4: every N-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an N-ary quasigroup can be represented as a composition of K-ary and (N-K+1)-ary quasigroups for some K from 2 to N-1, where the order of arguments in the representation can differ from the original order. The set of semilinear N-ary quasigroups has a characterization in terms of Boolean functions. Keywords: Latin hypercube, n-ary quasigroup, reducibilityComment: 10pp. V2: revise

On reconstructing reducible n-ary quasigroups and switching subquasigroups

(1) We prove that, provided n>=4, a permutably reducible n-ary quasigroup is uniquely specified by its values on the n-ples containing zero. (2) We observe that for each n,k>=2 and r<=[k/2] there exists a reducible n-ary quasigroup of order k with an n-ary subquasigroup of order r. As corollaries, we have the following: (3) For each k>=4 and n>=3 we can construct a permutably irreducible n-ary quasigroup of order k. (4) The number of n-ary quasigroups of order k>3 has double-exponential growth as n tends to infinity; it is greater than exp exp(n ln[k/3]) if k>=6, and exp exp(n (ln 3)/3 - 0.44) if k=5.Comment: 12pp. V.4: improved lower bound (last section), orders 5 and

An upper bound on the number of frequency hypercubes

A frequency $n$-cube $F^n(q;l_0,...,l_{m-1})$ is an $n$-dimensional $q$-by-...-by-$q$ array, where $q = l_0+...+l_{m-1}$, filled by numbers $0,...,m-1$ with the property that each line contains exactly $l_i$ cells with symbol $i$, $i = 0,...,m-1$ (a line consists of $q$ cells of the array differing in one coordinate). The trivial upper bound on the number of frequency $n$-cubes is $m^{(q-1)^{n}}$. We improve that lower bound for $n>2$, replacing $q-1$ by a smaller value, by constructing a testing set of size $s^{n}$, $s<q-1$, for frequency $n$-cubes (a testing sets is a collection of cells of an array the values in which uniquely determine the array with given parameters). We also construct new testing sets for generalized frequency $n$-cubes, which are essentially correlation-immune functions in $n$ $q$-valued arguments; the cardinalities of new testing sets are smaller than for testing sets known before. Keywords: frequency hypercube, correlation-immune function, latin hypercube, testing set

On the cardinality spectrum and the number of latin bitrades of order 3

By a (latin) unitrade, we call a set of vertices of the Hamming graph that is intersects with every maximal clique in $0$ or $2$ vertices. A bitrade is a bipartite unitrade, that is, a unitrade splittable into two independent sets. We study the cardinality spectrum of the bitrades in the Hamming graph $H(n,k)$ with $k=3$ (ternary hypercube) and the growth of the number of such bitrades as $n$ grows. In particular, we determine all possible (up to $2.5\cdot 2^n$) and large (from $14\cdot 3^{n-3}$) cardinatities of bitrades and prove that the cardinality of a bitrade is compartible to $0$ or $2^n$ modulo $3$ (this result has a treatment in terms of a ternary code of Reed--Muller type). A part of the results is valid for any $k$. We prove that the number of nonequivalent bitrades is not less than $2^{(2/3-o(1))n}$ and is not greater than $2^{\alpha^n}$, $\alpha<2$, as $n\to\infty$.Comment: 18 pp. In Russia

Etude de la formation d'image par auto-focalisation pour le développement d'un dispositif visuel non conventionnel de la réalité augmentée

Augmented Reality (AR) devices such as Near Eye Displays (NEDs) require both aesthetic design and high performance. Conventional NEDs are strongly limited by optical design constraints related to the basic principles of geometric optics. Our unconventional concept based on the self-focusing effect tries to overcome these constraints. The main idea of the self-focusing prototype is to emit directly the light field at the surface of the glass in a way the eye can form an image on the retina. It let us design a smart glasses prototype without an optical system between the viewer’s eye and the display. This ambition demands the association of technologies in integrated photonics, holography, and the implementation of diffractive effects. My research is focused on the theoretical description and experimental validation of the self-focusing effect for the NED-prototype image formation conceived in CEA-LETI. It is particularly oriented toward the understanding, simulation, and experimental evaluation of the relation between Emissive Points Distribution (EPD) and self-focused pattern (spel). Various types of EPD were studied and a feasible EPD was proposed. I proposed a metric to evaluate the quality of spel-formation and simulated self-focusing projection of our concept. The self-focusing performance of EPD was validated experimentally with a setup imitating our unconventional NED. In the second part of the work, I investigated the image formation capability of the self-focusing effect. Firstly, it was done with simulations based on a simplified spel-formation approach and then validated experimentally with a self-focusing image formation setup.Les dispositifs de Réalité Augmentée (RA) comme les écrans proches de l’œil (Near Eye Display ou NED) nécessitent à la fois une conception esthétique et des performances élevées. Les NED conventionnels sont très limités par les contraintes de conception liées aux principes de l’optique. Notre concept non conventionnel, basé sur l'effet d'autofocalisation, tente de surmonter ces contraintes. L'idée est d'émettre directement le champ lumineux à la surface du verre de manière à ce que l'œil puisse former une image sur la rétine. Cela permet de concevoir des lunettes sans système optique entre l'œil et l'écran. Cette ambition nécessite d'associer les technologies de la photonique intégrée, de l'holographie, et de la mise en œuvre d'effets diffractifs. Ma thèse porte sur la description théorique et la validation expérimentale de l'effet d'autofocalisation dans le prototype NED conçu au CEA-LETI. Elle est orientée vers la compréhension, la simulation et l'évaluation expérimentale de la relation entre la distribution des points émissifs (EPD) et le motif autofocus (spel). Différents types d'EPD ont été étudiés et une EPD réalisable a été identifiée. J'ai proposé une métrique pour évaluer la qualité de la formation du spel et simulé une projection autofocalisée. Les performances de cette l'EPD ont été validées expérimentalement avec une configuration imitant notre NED non conventionnel. Dans la deuxième partie de ce travail, j'ai étudié la capacité de formation d'images de l'effet d'autofocalisation, à l'aide de simulations basées sur une approche simplifiée de la formation de spels, puis d’une validation expérimentale avec une configuration de formation d'images autofocus

Etude de la formation d'image par auto-focalisation pour le développement d'un dispositif visuel non conventionnel de la réalité augmentée

Augmented Reality (AR) devices such as Near Eye Displays (NEDs) require both aesthetic design and high performance. Conventional NEDs are strongly limited by optical design constraints related to the basic principles of geometric optics. Our unconventional concept based on the self-focusing effect tries to overcome these constraints. The main idea of the self-focusing prototype is to emit directly the light field at the surface of the glass in a way the eye can form an image on the retina. It let us design a smart glasses prototype without an optical system between the viewer’s eye and the display. This ambition demands the association of technologies in integrated photonics, holography, and the implementation of diffractive effects. My research is focused on the theoretical description and experimental validation of the self-focusing effect for the NED-prototype image formation conceived in CEA-LETI. It is particularly oriented toward the understanding, simulation, and experimental evaluation of the relation between Emissive Points Distribution (EPD) and self-focused pattern (spel). Various types of EPD were studied and a feasible EPD was proposed. I proposed a metric to evaluate the quality of spel-formation and simulated self-focusing projection of our concept. The self-focusing performance of EPD was validated experimentally with a setup imitating our unconventional NED. In the second part of the work, I investigated the image formation capability of the self-focusing effect. Firstly, it was done with simulations based on a simplified spel-formation approach and then validated experimentally with a self-focusing image formation setup.Les dispositifs de Réalité Augmentée (RA) comme les écrans proches de l’œil (Near Eye Display ou NED) nécessitent à la fois une conception esthétique et des performances élevées. Les NED conventionnels sont très limités par les contraintes de conception liées aux principes de l’optique. Notre concept non conventionnel, basé sur l'effet d'autofocalisation, tente de surmonter ces contraintes. L'idée est d'émettre directement le champ lumineux à la surface du verre de manière à ce que l'œil puisse former une image sur la rétine. Cela permet de concevoir des lunettes sans système optique entre l'œil et l'écran. Cette ambition nécessite d'associer les technologies de la photonique intégrée, de l'holographie, et de la mise en œuvre d'effets diffractifs. Ma thèse porte sur la description théorique et la validation expérimentale de l'effet d'autofocalisation dans le prototype NED conçu au CEA-LETI. Elle est orientée vers la compréhension, la simulation et l'évaluation expérimentale de la relation entre la distribution des points émissifs (EPD) et le motif autofocus (spel). Différents types d'EPD ont été étudiés et une EPD réalisable a été identifiée. J'ai proposé une métrique pour évaluer la qualité de la formation du spel et simulé une projection autofocalisée. Les performances de cette l'EPD ont été validées expérimentalement avec une configuration imitant notre NED non conventionnel. Dans la deuxième partie de ce travail, j'ai étudié la capacité de formation d'images de l'effet d'autofocalisation, à l'aide de simulations basées sur une approche simplifiée de la formation de spels, puis d’une validation expérimentale avec une configuration de formation d'images autofocus