Transitive distance-regular graphs from linear groups $L(۳,q)$‎, ‎$q = ۲,۳,۴,۵$

In this paper we classify distance-regular graphs‎, ‎including strongly regular graphs‎, ‎admitting a transitive action of the linear groups $L(۳,۲)$‎, ‎$L(۳,۳)$‎, ‎$L(۳,۴)$ and $L(۳,۵)$ for which the rank of the permutation representation is at most ۱۵‎. ‎We give details about constructed graphs‎. ‎In addition‎, ‎we construct self-orthogonal codes from distance-regular graphs obtained in this paper‎.

Keywords:

strongly regular graph , distance-regular graph , linear group , self-orthogonal code