Constructing commutative semifields of square order

一类新的2参数半域

A new family of semifields with 2 parameters

给出了一类新的双参数交换半域。它的左核和中核都是确定的。此外，我们证明了，对于任何不同的参数对，这些半域是不是同位素。对于某些特殊的参数，族中的一个半域可以导致两个不等价的平面函数。最后，利用类似的构造，给出了新的APN函数。

半域、相对差集和Bent函数

Semifields, relative difference sets, and bent functions

最近，由于一些新族的发现和分类问题的进展，人们对半域的兴趣增加了。可换半域是一类重要的平面函数，它等价于某些平面函数（在奇特征的情况下）和修正的平面函数（在偶特征的情况下）。类似地，交换半域等价于相对差集。本调查的目的是描述这些概念之间的联系。此外，我们将讨论幂映射是平面的，并考虑平面映射的分量函数，它也可以被看作是相对差集的投影。结果表明，在偶特征的情况下，分量函数与negabent函数以及$\mathbb{Z}_4$值bent函数有关。

Symplectic 4-dimensional semifields of order $8^4$ and $9^4$

We classify symplectic 4-dimensional semifields over $\mathbb{F}_q$, for $q\leq 9$, thereby extending (and confirming) the previously obtained classifications for $q\leq 7$. The classification is obtained by classifying all symplectic semifield subspaces in $\mathrm{PG}(9,q)$ for $q\leq 9$ up to $K$-equivalence, where $K\leq \mathrm{PGL}(10,q)$ is the lift of $\mathrm{PGL}(4,q)$ under the Veronese embedding of $\mathrm{PG}(3,q)$ in $\mathrm{PG}(9,q)$ of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for $q$ even, $q\leq 8$. For $q$ odd, a…

A family of semifields in characteristic 2

We construct and describe the basic properties of a family of semifields in characteristic $2.$ The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nucleus. The non-degenerate members of our family display a very different behaviour. Their left and right nucleus agrees with the center, the middle nucleus is quadratic over the center. None…

关于交换预半群的合痕与强合痕

On isotopisms and strong isotopisms of commutative presemifields

本文证明了Bierbrauer在\cite{BierbrauerSub}中构造的$P(q,\ell)$（$q$奇素幂和$\ell>1$奇）交换半域与Budaghyan和Helleseth在\cite{BuHe2008}中构造的某些交换预半群是同位素的.证明了它们是强同位素的当且仅当$q\equiv 1(mod\,4)$。因此，对于每个$q\equiv -1(mod\,4)$，存在定义CCZ-不等平面DO多项式的$q^{2\ell}$（$\ell>1$奇）阶的同位素交换预半群.

辛展开、平面函数和互正交基

Symplectic spreads, planar functions and mutually unbiased bases

本文通过辛展开和伪平面函数提出新的方法，构造了特殊李代数的正交分解和互正交基的完整集合。

The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra

We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show …

构造厚B_h集

Constructing Thick $B_h$-sets

本文推广了Bose-Chowla和Singer的构造方法，研究了Z中B_h集的直径界限，并提出了若干开放问题。

关于特征三平面函数的一个分类

On a classification of planar functions in characteristic three

本文系统研究了特征三有限域上平面函数的CCZ-等价与同位等价分类，完全解决了$n\le 11$时已知平面函数的同位等价是否产生新CCZ-类的问题，并在$\mathbb{F}_{3^6}$和$\mathbb{F}_{3^9}$上发现了九个新的零散平面函数。

复合偏序集的默比乌斯函数的合理性

Rationality of the Möbius function of a composition poset

我们考虑zeta和默比乌斯函数的偏序的整数组成首先研究了贝杰龙，Bousquet-Mélou，和Dulucq。这个偏序集的默比乌斯函数是由萨根和瓦特确定的。我们证明了各种形式的幂级数的合理性的非交换变量的系数的zeta函数和默比乌斯函数的评价。这些证明可以直接从定义中得到，也可以通过构造有限状态自动机得到。我们还获得了明确的表达式，通过专门的变量交换的生成函数。我们重新证明萨根和瓦特的公式的默比乌斯函数使用这种机制。这些结果是密切相关的Björner和Bjötenauer关于子词序，我们讨论了一个共同的推广。