标题： 代码和伪几何设计从三元$m$序列与 Welch-type 间隔$d=2\cdot 3^{(n-1)/2}+1$ 显示英文标题

标题： Codes and Pseudo-Geometric Designs from the Ternary $m$-Sequences with Welch-type decimation $d=2\cdot 3^{(n-1)/2}+1$

摘要： 伪几何设计是组合设计，其参数与有限几何设计相同，但不与该设计同构。 据我们所知，许多伪几何设计已通过有限几何和组合学的方法构造出来。 然而，目前还没有利用编码理论方法构造出参数为$S\left (2, q+1,(q^n-1)/(q-1)\right )$的伪几何设计。 在本文中，我们使用循环码来构造伪几何设计。 我们首先从具有 Welch 型分量$d=2\cdot 3^{(n-1)/2}+1$的$m$序列中提出一族三元循环码，并利用这些循环码及其对偶码得到一些无限族的 2-设计和一族 Steiner 系统$S\left (2, 4, (3^n-1)/2\right )$。 此外，还确定了这些循环码及其缩短码的参数。 其中一些三元码是最佳或几乎最佳的。 最后，我们证明其中的一个所得的 Steiner 系统与射影空间$\mathrm{PG}(n-1,3)$的点线设计不等价，因此是一个伪几何设计。 显示英文摘要

摘要： Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but which are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters $S\left (2, q+1,(q^n-1)/(q-1)\right )$ is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the $m$-sequences with Welch-type decimation $d=2\cdot 3^{(n-1)/2}+1$, and obtain some infinite family of 2-designs and a family of Steiner systems $S\left (2, 4, (3^n-1)/2\right )$ using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space $\mathrm{PG}(n-1,3)$ and thus is a pseudo-geometric design.