# (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.

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I may not be up to speed with coding theory, but I may still not be the only member who might be able to answer, but does not know what the heck is S18? I don't think it is a standard name of any code. – Jyrki Lahtonen Jun 10 '13 at 10:32

## 1 Answer

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.

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I thought S_18 must be special group, because it is always said this code is equivalent to S_18, nothing else. And I read all the paper you mentioned. And then I found the generator vector of S_18 in the paper "Classiﬁcation of Generalized Hadamard Matrices H(6,3) and Quaternary Hermitian Self-Dual Codes of Length 18". Thanks. – pinarcomak Jun 12 '13 at 5:33
In this context, $S_{18}$ cannot be a group. If a code is equivalent to something, then something must be a code, too. – azimut Jun 12 '13 at 14:35
+1 Good work digging all that up! – Jyrki Lahtonen Jun 12 '13 at 18:39
@JyrkiLahtonen: Thank you! – azimut Jun 13 '13 at 9:57