Related papers: On transitive uniform partitions of F^n into binar…

A vector space partition of $\mathbb{F}_q^v$ is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring…

Let $n \in \mathbb{N}_{\geq 2}$. We prove that for every $k \geq 4$ there exist uniform but non-homogeneous Steiner bundles on $\mathbb{P}^n$ of $k$-type with disconnected splitting type, and we further investigate almost-uniform Steiner…

Pattern avoidance in the symmetric group $S_n$ has provided a number of useful connections between seemingly unrelated problems from stack-sorting to Schubert varieties. Recent work has generalized these results to $S_n\wr C_c$, the objects…

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power…

Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet…

The $\ell$-Galois hull $h_{\ell}(C)$ of an $[n,k]$ linear code $C$ over a finite field $\mathbb{F}_q$ is the intersection of $C$ and $C^{{\bot}_{\ell}}$, where $C^{\bot_{\ell}}$ denotes the $\ell$-Galois dual of $C$ which introduced by Fan…

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher's theorem: for all $m \geq 2$ and…

We study the Baire class one countable colorings, i.e., the countable partitions into $F_\sigma$ sets. Such a partition gives a covering of the diagonal into countably many $F_\sigma$ squares. This leads to the study of countable unions of…

We propose conjectural semiorthogonal decompositions for Fano schemes of linear subspaces on intersections of two quadrics, in terms of symmetric powers of the associated hyperelliptic (resp. stacky) curve. When the intersection is…

In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…

A \emph{set partition} of the set $[n]=\{1,...c,n\}$ is a collection of disjoint blocks $B_1,B_2,...c, B_d$ whose union is $[n]$. We choose the ordering of the blocks so that they satisfy $\min B_1<\min B_2<...b<\min B_d$. We represent such…

In the composite fermion model of the fractional quantum Hall effect, composite fermions experience, in addition to the usual potential disorder, also a magnetic flux disorder. Motivated by this, we investigate the localization properties…

A code is called propelinear if its automorphism group contains a subgroup that acts regularly on its codewords, which is called a propelinear structure on the code. In the paper a classification of the propelinear structures on the…

We consider a \emph{code} to be a subset of the vertex set of a \emph{Hamming graph}. In this setting a \emph{neighbour} of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the…

We continue the study of the class of binary extended perfect propelinear codes constructed in the previous paper and consider their permutation automorphism (symmetry) groups and Steiner quadruple systems. We show that the automorphism…

An affine vector space partition of $\operatorname{AG}(n,q)$ is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for…

In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set…

Let $H$ be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let $f \in H$ be a subpolynomial function. Let $\mathcal{P} = \{2, 3, 5, 7, \dots \}$ be the (naturally…

We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where…

We solve several first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs $H(n,q)$. The most uplifting result is the existence of a $\{13,6,1;1,6,9\}$-CR code in $H(n,2)$, $n\ge 13$. We also…