On the Dimensions of Certain LDPC Codes Based on$q$-Regular Bipartite Graphs

An explicit construction of a family of binary low-density parity check (LDPC) codes called
$hbox LU(3,q)$
, where
$q$
is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when
$q$
is odd. The conjecture is proved in this note. The proof involves the geometry of a four-dimensional (4-D) symplectic vector space and the action of the symplectic group and its subgroups.