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.

翻译：伪几何设计是一类与有限几何设计具有相同参数但不同构于该设计的组合设计。据我们所知，已有许多伪几何设计通过有限几何与组合方法构造。然而，至今尚未有参数为 $S\left (2, q+1,(q^n-1)/(q-1)\right )$ 的伪几何设计通过编码理论方法构造。本文利用循环码构造伪几何设计。首先，我们从具有Welch型抽取 $d=2\cdot 3^{(n-1)/2}+1$ 的 $m$ 序列出发，构造一族三元循环码，并利用这些循环码及其对偶码得到若干无限族2-设计与一族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，其中部分三元码为最优或接近最优。最后，我们证明其中一个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。