Pseudo-geometric designs are designs which share the same parameters as a finite geometry design, but which are not isomorphic to that design. As far as we know, only a very small number of pseudo-geometric designs have been constructed and no pseudo-geometric designs with the same parameters $S\left (2, q+1,(q^n-1)/(q-1)\right )$ as the point-line designs of the projective spaces $\mathrm{PG}(n-1,q)$ were found. In this paper, we present a family of ternary cyclic codes from the $m$-sequences with Welch-type decimation $d=2\cdot 3^{(n-1)/2}+1$, and construct 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. We show that one of these 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. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal.

翻译：伪几何设计是指与有限几何设计具有相同参数、但不同构于该几何设计的设计。据我们所知，仅有极少数伪几何设计被构造出来，且尚未发现与射影空间$\mathrm{PG}(n-1,q)$的点线设计具有相同参数$S\left (2, q+1,(q^n-1)/(q-1)\right )$的伪几何设计。本文从具有Welch型抽取$d=2\cdot 3^{(n-1)/2}+1$的$m$序列出发，给出了一族三元循环码，并利用这些循环码及其对偶码构造了若干无限族2-设计和一族Steiner系统$S\left (2, 4, (3^n-1)/2\right )$。我们证明其中一个Steiner系统不等价于射影空间$\mathrm{PG}(n-1,3)$的点线设计，因此是一个伪几何设计。此外，还确定了这些循环码及其缩短码的参数，其中部分三元码是最优或接近最优的。