论文标题
有限田的紧密heffter阵列
Tight Heffter arrays from finite fields
论文作者
论文摘要
在将紧密的heffter阵列H $（m，n）$扩展到任何订单$ 200万美元+1 $的经典概念之后，我们为小学的Abelian Abelian tightheffter阵列提供了直接的构造，因此尤其是针对Prime紧密的Heffter Arrays。如果$ q = 2Mn+1 $是一种主要力量，我们说一个小学的Abelian H $（m，n）$是``超过$ \ m \ m \ m \ m \ m \ m {f} _q $''，因为它的构造，我们在许多情况下都在许多情况下，在许多情况下，我们证明了$ q $的添加性和多样性的结构。 say $A$, can be obtained very easily by imposing that $A$ has rank 1 and, possibly, a rich group of {\it multipliers}, that are elements $u$ of $\mathbb{F}_q$ such that $u A=A$ up to a permutation of rows and columns. An H$(m,n)$ over $\mathbb{F}_q$ will be said {\it如果其乘数组的顺序是$ m $和$ n $的奇数零件的最低倍数，因为这是最大可能的订单。
After extending the classic notion of a tight Heffter array H$(m,n)$ to any group of order $2mn+1$, we give direct constructions for elementary abelian tight Heffter arrays, hence in particular for prime tight Heffter arrays. If $q=2mn+1$ is a prime power, we say that an elementary abelian H$(m,n)$ is ``over $\mathbb{F}_q$" since, for its construction, we exploit both the additive and multiplicative structure of the field of order $q$. We show that in many cases a direct construction of an H$(m,n)$ over $\mathbb{F}_q$, say $A$, can be obtained very easily by imposing that $A$ has rank 1 and, possibly, a rich group of {\it multipliers}, that are elements $u$ of $\mathbb{F}_q$ such that $u A=A$ up to a permutation of rows and columns. An H$(m,n)$ over $\mathbb{F}_q$ will be said {\it optimal} if the order of its group of multipliers is the least common multiple of the odd parts of $m$ and $n$, since this is the maximum possible order for it. The main result is an explicit construction of a rank-one H$(m,n)$ -- reaching almost always the optimality -- for all admissible pairs $(m,n)$ for which there exist two distinct odd primes $p$, $p'$ dividing $m$ and $n$, respectively.
