Common Information, Matroid Representation, and Secret Sharing for Matroid Ports

拟阵熵函数：构造、刻画与表示

Matroidal Entropy Functions: Constructions, Characterizations and Representations

该研究通过构造拟阵运算的变强度阵列，刻画了正则拟阵及部分与均匀拟阵U_{2,4}同p特征集的拟阵诱导的拟阵熵函数，为信息论相关问题的容量刻画与码构造提供支撑。

格路径拟阵的高效表示

Efficient Representation of Lattice Path Matroids

本文基于分层秘密共享方案、分层拟阵端口刻画及多项式规模隔离权函数，提出了在有限域上高效构造格路径拟阵表示的确定性算法。

Ported Tutte Functions of Extensors and Oriented Matroids

The Tutte equations are ported (or set-pointed) when the equations F(N) = g_e F(N/e) + r_e F(N\e) are omitted for elements e in a distinguished set called ports. Solutions F can distinguish different orientations of the same matroid. A ported extensor with ground set is a decomposible element in the exterior algebra over a vector space with a given basis, called the ground set, containing a distinguished subset called ports. These can represent ported matroids and have analogous dualization, deletion and contraction operations. A ported extensor function is defined using dualization, port elem…

On the representability of the bi-uniform matroid

Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given bi-uniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all bi-uniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of bi-uniform matroids, while the known genera…

Quantum Codes and Symplectic Matroids

The correspondence between linear codes and representable matroids is well known. But a similar correspondence between quantum codes and matroids is not known. We show that representable symplectic matroids over a finite field $\mathbb{F}_q$ correspond to $\mathbb{F}_q$-linear quantum codes. Although this connection is straightforward, it does not appear to have been made earlier in literature. The correspondence is made through isotropic subspaces. We also show that the popular Calderbank-Shor-Steane (CSS) codes are essentially the homogenous symplectic matroids while the graph states, which …

定向拟阵上的模式识别：超立方体图中的对称循环。VI

Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. VI

本文探讨二元数组精确模型表示的线性代数、组合与应用层面，并以字符串的两幅线性代数画像加以说明。

几乎仿射向量秩度量码

Almost Affine Vector Rank-Metric Codes

本文定义几乎仿射向量秩度量码，证明其自然诱导q-拟阵，并建立穿孔和缩短操作与q-拟阵的限制和收缩的对应关系，同时证明秩权重和形式对偶距离分布由诱导的q-拟阵决定，最后给出严格几乎仿射秩度量码的构造实例。

Some inequalities for the Tutte polynomial

We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.

Implicit Linear Algebra and Basic Circuit Theory II: port behaviour of rigid multiports

In this paper, we define the notion of rigidity for linear electrical multiports and for matroid pairs. We show the parallel between the two and study the consequences of this parallel. We present applications to testing, using purely matroidal methods, whether a connection of rigid multiports yields a linear network with unique solution. We also indicate that rigidity can be regarded as the closest notion to duality that can be hoped for, when the spaces correspond to different physical constraints, such as topological and device characteristic. A multiport is an ordered pair $(\V^1_{AB},\A^2…

折叠代数拟阵：特征刚性与几乎熵分离

Folded-Algebraic Matroids: Characteristic Rigidity and Almost-Entropic Separation

该研究引入折叠代数拟阵，证明其与几乎熵拟阵为真包含关系，构造出几乎熵但非折叠代数的拟阵，还得到具有不相容特征要求的连通端口，明确了相关表示类的分离区域。