(18,9,8) self-dual quaternary codes vs S18 i am wondering about the form of S18. It is written that [18,9,8] self-dual quaternary codes is equivalent to S18. there is a generator matrix of this quaternary code, ok, but how it can be equivalent to S18? S18 is the symmetric group. S_{18} maybe better.

The code $S_{18}$ was defined in 1978 in the article F. J. MacWilliams; A. M. Odlyzko, N. J. A. Sloane, H. N. Ward: Self-dual codes over $GF(4)$ as an extended cyclic code, see page 310.

In 1997, in W. C. Huffman: Characterization of quaternary extremal codes of length $18$ and $20$ it was shown that up to equivalence, $S_{18}$ is unique among the extremal Hermitian self-dual $\mathbb F_4$-linear $[18,9,8]$ codes. The proof involves computer results. This should answer your question, I guess.

As a side note: In the article C. Bachoc; P. Gaborit: On extremal additive $\mathbb{F}_4$ codes of length 10 to 18, it is shown that every extremal even self-dual $\mathbb F_4$-additive code with some additional property (in the article, $s(C) = 0$) is equivalent to $S_{18}$. This article was written in 2000. I don't know if there was any progress in the meantime.