We consider the induced subgraph of the corresponding Grassmann graph formed by q-ary simplex codes of dimension $2$, q≥ 5. This graph contains precisely two types of maximal cliques. If $q=5$, then for any two maximal cliques of the same type there is a monomial linear automorphism transferring one of them to the other. Examples concerning the cases $q=7,11$ finish the note.
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Kwiatkowski et al. (2024) studied this question.
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