Infinite families of 2-designs from GA_1(q) actions
Infinite families of 2-designs from GA_1(q) actions
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发表时间：
2017-07
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通讯作者：
Hao Liu;C. Ding
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作者：
Hao Liu;C. Ding
团体行动是获得 $t$ 设计的标准方法。在这种方法中，选择具有一定程度的传递性或同质性的特定置换群以及一组适当的基块对于获得具有可计算参数$t、v、k$和$\lambda$的$t$-$(v,k,\lambda)$设计非常重要。一般仿射群$\GA_1(q)$在$\gf(q)$上是$2$传递的，并且具有相对较小的尺寸。在本文中，我们确定了从组 $\GA_1(q)$ 在某些基本块上的作用获得的许多 $2$-设计的无限族的参数，并证明一些 $2$-设计产生具有已知最优或最佳参数的线性代码。还提出了未解决的问题。
Group action is a standard approach to obtain $t$-designs. In this approach, selecting a specific permutation group with a certain degree of transitivity or homogeneity and a proper set of base blocks is important for obtaining $t$-$(v, k, \lambda)$ designs with computable parameters $t, v, k$, and $\lambda$. The general affine group $\GA_1(q)$ is $2$-transitive on $\gf(q)$, and has relatively a small size. In this paper, we determine the parameters of a number of infinite families of $2$-designs obtained from the action of the group $\GA_1(q)$ on certain base blocks, and demonstrate that some of the $2$-designs give rise to linear codes with optimal or best parameters known. Open problems are also presented.