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

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.