The doubly truncated Witt graph is the graph on 330 vertices related to a 3-
design (Brouwer et al. 1989, p. 367).

The doubly truncated Witt graph can be constructed by removing all vectors of the large Witt design containing any two arbitrarily chosen symbols. Consider the 759
vertices of the large Witt graph as words of weight
8 in extended binary Golay. Call the octads, and view them as sets of size 8. Pick
one coordinate position. 253 octads have a 1 there, 506 octads have a 0 there. Pick
a second coordinate position. Of the 506, there are 176 with a 1 there, and 330 with
a 0. The induced subgraph from 759, or from 506 on this 330 gives the doubly truncated
Witt graph (A. E. Brouwer, pers. comm., Jun. 8, 2009).

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "The Doubly Truncated Witt Graph Associated to ." §11.4C in Distance
Regular Graphs. New York: Springer-Verlag, pp. 211 and 368-369, 1989.DistanceRegular.org.
"Doubly Truncated Witt Graph." http://www.distanceregular.org/graphs/dtwitt.html.van
Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their
Spectrum?" Lin. Algebra Appl.373, 139-162, 2003.