Gleason's Theorem on Self-Dual Codes and Its Generalizations

A generalization of a result of Dong and Santos-Sturmfels on the Alexander dual of spheres and balls

We prove a generalization of a result by Dong and Santos-Sturmfels about the homotopy type of the Alexander dual of balls and spheres. Our results involve NH-manifolds, which were recently introduced as the non-homogeneous (or non-pure) counterpart of classical polyhedral manifolds. We show that the Alexander dual of an NH-ball is contractible and the Alexander dual of an NH-sphere is homotopy equivalent to a sphere. We also prove that NH-balls and NH-spheres arise naturally as the double duals of standard balls and spheres.

全对偶整数性的推广

Generalizations of Total Dual Integrality

本文通过设计新工具研究全对偶整数性的变体，在非退化多面体情形下给出其几何刻画，并证明 Seymour 二进猜想的新特殊情形。

Characterization of regular checkerboard colourable twisted duals of ribbon graphs

The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. The twisted dual is a further generalization by combining the partial Petrial. Given a ribbon graph $G$, in this paper, we first characterize regular partial duals of the ribbon graph $G$ by using spanning quasi-tree and its related shorter marking arrow sequence set. Then we chara…

New binary self-dual codes via a generalization of the four circulant construction

In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets F_2, F_2 + uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.

On Generalizations of the Pell Sequence

In this paper, we investigate the generalized Pell sequence, the generalized complex Pell sequence and the generalized dual Pell sequence using the Pell numbers. We obtain special cases of these sequences. Furthermore, we give recurrence relations, vectors, the silver ratio and Binet's formula for the above-investigated sequences.

Percolation on self-dual polygon configurations

Recently, Scullard and Ziff noticed that a broad class of planar percolation models are self-dual under a simple condition that, in a parametrized version of such a model, reduces to a single equation. They state that the solution of the resulting equation gives the critical point. However, just as in the classical case of bond percolation on the square lattice, self-duality is simply the starting point: the mathematical difficulty is precisely showing that self-duality implies criticality. Here we do so for a generalization of the models considered by Scullard and Ziff. In these models, the s…

On $Z_{2^k}$-Dual Binary Codes

A new generalization of the Gray map is introduced. The new generalization $Φ: Z_{2^k}^n \to Z_{2}^{2^{k-1}n}$ is connected with the known generalized Gray map $ϕ$ in the following way: if we take two dual linear $Z_{2^k}$-codes and construct binary codes from them using the generalizations $ϕ$ and $Φ$ of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of $Z_{2^k}$-linear Hadamard codes and co-$Z_{2^k}$-linear extended 1-perfect codes are described, where co-$Z_{2^k}$-linearity means that the code can be obtained from a …

对偶Littlewood恒等式的测度与推广

Measures and generalizations of dual Littlewood identities

该研究通过引入特定向量族，以新方式推导B、C、D型对偶Littlewood恒等式，获自由费米子构造证明相关测度是行列式形式，还建立广义Littlewood恒等式，包含著名恒等式特殊情况并给出李超代数恒等式新证。