Some optimal codes related to graphs invariant under the alternating group $A_8$

The alternating group $A_8$, acts as a primitive rank-3 group of degree $35$ on the set of lines of $V_4(2)$ with line stabilizer isomorphic to $2^4:(S_3 \times S_3)$ and orbits of lengths 1, 16 and 18 respectively. This action defines the unique strongly regular $(35, 16, 6, 8)$ graph. The paper examines the binary (resp. ternary) codes spanned by the rows of this graph, and its complement. We establish some properties of the codes and use the geometry of the designs and graphs to give an account on the nature of some classes of codewords, in particular those of minimum weight. Further, we show that the codes with parameters $[35, 28, 4]_2,[35, 6, 16]_2,[35, 29, 3]_2,[28, 7, 12]_2,[28, 21,4]_2, [36, 7, 16]_2, [36,29,4]_2$ and $[64, 56, 4]_2$ are all optimal. In addition, we show that the codes with parameters $[35, 13, 12]_3, [35, 22, 5]_3,[35, 14, 11]_3, [35, 21, 6]_3$ are all near-optimal for the given length and dimension.