Mariusz Kwiatkowski, Mark Pankov
We give a complete description of the distance relation on the graph of $4$-ary simplex codes of dimension $2$. This is a connected graph of diameter $3$. For every vertex we determine the sets of all vertices at distance $i\in\{1,2,3\}$ and describe their symmetries.
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On maximal cliques in the graph of simplex codes
本文研究单纯码在 Grassmann 图中形成的诱导子图，刻画其两类极大团，并证明第一类团可由单项式线性自同构相互转移，第二类团结构更复杂且大小可变。
Integer points in a simplex and related Diophantine problems: Hardy--Littlewood asymptotics in higher dimensions
本文扩展了哈代和利特尔伍德在二维右三角形中整数点计数的结果到高维空间，探讨单纯形中的整数点分布及相关丢番图问题。
Maximal cliques in the graph of $5$-ary simplex codes of dimension two
本文研究二维$q$元单纯形码构成的格拉斯曼诱导子图，证明其恰有两类最大团，且$q=5$时同类型最大团可由单项线性自同构互变，并给出$q=7,11$的示例。
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first and second order, ordinary, linear differential equations. Regarding the first class, the corresponding identities amount to a proof of the exponential formula of labelled counting. The identities in the second class can be used to establish certain geometric properties of the simplex of bounded, ordered, integer tuples. We present three theorems that suppo…
This paper presents a new alphabet-dependent bound for codes with hierarchical locality. Then, the complete list of possible localities is derived for a class of codes obtained by deleting specific columns from a Simplex code. This list is used to show that these codes are optimal codes with hierarchical locality.
Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4$
本文研究了$n$元集合的$2m$元子集点线几何中，对应于二元单纯码及对称$(n,2m,m)$-设计的极大团，并在$k=4$时为五个著名的对称$(15,8,4)$-设计提供了几何解释。
A note on the diameter of graphs of two-dimensional simplex codes
本文证明了二维q元单纯形码诱导的Grassmann子图直径对所有素数幂q均为3（q=2,3例外），从而完全确定了该直径，方法结合桥引理、计数论证与矩计算。
The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a simple, constructive proof of a result that is "in between" these two properties. Our approach is to relate these properties to certain (old and new) additive problems in finite abelian groups. We also formulate a conjecture for finite abelian groups that generalizes the above-mentioned conjecture.
We present equivalent formulations for concepts related to set families for which every subfamily with empty intersection has a bounded sub-collection with empty intersection. Hereby, we summarize the progress on the related questions about the maximum size of such families. In this work we solve a boundary case of a problem of Tuza for non-trivial $q$-Helly families, by applying Karamata's inequality and determining the minimum size of a $2$-self-centered graph for which the common neighborhood of every pair of vertices contains a clique of size $q-2$.
Consider the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the field of $q$ elements ($1<k<n-1$) and denote by $Π(n,k)_q$ the restriction of this graph to the set of projective $[n,k]_q$ codes. In the case when $q\ge \binom{n}{2}$, we show that the graph $Π(n,k)_q$ is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simp…
The paper deals with a three-dimensional family of diffusion processes on an infinite-dimensional simplex. These processes were constructed by Borodin and Olshanski (arXiv:0706.1034; arXiv:0902.3395), and they include, as limit objects, the Ethier-Kurtz's infinitely-many-neutral-allels diffusion model (1981) and its extension found by Petrov (arXiv:0708.1930). Each process X from our family possesses a symmetrising measure M. Our main result is that the transition function of X has continuous density with respect to M. This is a generalization of earlier results due to Ethier (1992) and to Fen…
Smith Normal Forms of Graphical Hermite Simplices
引入图形埃尔米特单纯形族，研究其顶点向量矩阵的史密斯标准型，等价于研究这些矩阵的余核群结构，动机是研究顶点受小扰动后格单纯形的行为。
Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brains sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions - in contrast to Erdős-Rényi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices. We present a new method capable of deliver…
We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a finite projective or affine space. As a result from the geometric method we use for the weight enumeration, we also completely determine the set of supports of subcodes and words in an extension code.
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