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Pseudo-geometric designs are combinatorial designs which share the sameparameters as a finite geometry design, but which are not isomorphic to thatdesign. As far as we know, many pseudo-geometric designs have been constructedby the methods of finite geometries and combinatorics. However, none ofpseudo-geometric designs with the parameters $S\left (2,q+1,(q^n-1)/(q-1)\right )$ is constructed by the approach of coding theory. Inthis paper, we use cyclic codes to construct pseudo-geometric designs. Wefirstly present a family of ternary cyclic codes from the $m$-sequences withWelch-type decimation $d=2\cdot 3^{(n-1)/2}+1$, and obtain some infinite familyof 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 thesecyclic codes and their shortened codes are also determined. Some of thoseternary codes are optimal or almost optimal. Finally, we show that one of theseobtained Steiner systems is inequivalent to the point-line design of theprojective space $\mathrm{PG}(n-1,3)$ and thus is a pseudo-geometric design.

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