相关论文: Residual $q$-Fano Planes and Related Structures
In this paper we give the first construction of a q-analog of a Steiner system. Using a computer search we found at least 26 q-Steiner Systems S_2[2,3,13] admitting the normalizer of a singer cycle as a group of automorphisms.
Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…
The Fano resonance has been a familiar and important feature in atomic and molecular physics for more than half a century. Typically, the combination of a discrete state with one or more continua results in an asymmetric peak in the…
A $t\text{-}(n,K,\lambda;q)$ design, also called the $q$-analog of a $t$-wise balanced design, is a set ${\mathcal B}$ of subspaces with dimensions contained in $K$ of the $n$-dimensional vector space ${\mathbb F}_q^n$ over the finite field…
A $t$-$(n,k,\lambda)$ design over $\F_q$ is a collection of $k$-dimensional subspaces of $\F_q^n$, called blocks, such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over $\F_q$ are…
The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and…
It is shown that the automorphism group of a binary $q$-analog of the Fano plane is either trivial or of order $2$.
Fano mechanism is the universal explanation of asymmetric resonance appearing in different systems. We report the evidence of Fano-like resonance in selective reflection from a resonant two-level medium. We draw an analogy with the…
Finite projective geometries, especially the Fano plane, have been observed to arise in the context of certain quantum gate operators. We use Clifford algebras to explain why these geometries, both planar and higher dimensional, appear in…
An automorphism group of an incidence structure I induces a tactical decomposition on I. It is well known that tactical decompositions of t-designs satisfy certain necessary conditions which can be expressed as equations in terms of the…
Classical block designs are important combinatorial structures with a wide range of applications in Computer Science and Statistics. Here we give a new abstract description of block designs based on the arrow category construction. We show…
The classical problem of whether $m$th-powers with or without zero in a finite field $\mathbb{F}_q$ form a difference set has been extensively studied, and is related to many topics, such as flag transitive finite projective planes. In this…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
Coding theory and combinatorial $t$-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some…
An {\em $m\times n$ row-column factorial design} is an arrangement of the elements of a factorial design into a rectangular array. Such an array is used in experimental design, where the rows and columns can act as blocking factors. If for…
A q-design with parameters t-(v,k,lambda_t)_q is a pair (V, B) of the v-dimensional vector space V over GF(q) and a collection B of k-dimensional subspaces of V, such that each t-dimensional subspace of V is contained in precisely lambda_t…
Combinatorial designs have been studied for nearly 200 years. Fifty years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined…
A system of interacting classical oscillators is discussed, similar to a quantum mechanical system of a discrete energy level, interacting with the energy quasi-continuum of states considered Fano. The limit of a continuous spectrum is…
In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25.. There are several planes for which we have not found any unital. One or more than one…
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the $q$-analogues of projective systems and blocking sets. We also illustrate how to associate a classical…