For 4-ary simplex codes of dimension 2, the graph of codes is connected of diameter 3, with 162 vertices, each having 25 neighbors, 130 second neighbors, and 6 third neighbors, and the distance-2 layer splits into orbits of sizes 20, 90, and 20.
Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes , Ars Combin
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Graphs related to $2$-dimensional simplex codes
For 4-ary simplex codes of dimension 2, the graph of codes is connected of diameter 3, with 162 vertices, each having 25 neighbors, 130 second neighbors, and 6 third neighbors, and the distance-2 layer splits into orbits of sizes 20, 90, and 20.