  In view of the result of Kontsevich, now often called ``the fundamental
theorem of Vassiliev theory'', identifying the graded dual of the associated
graded vector space to the space of Vassiliev invariants filtered by degree
with the linear span of chord diagrams modulo the ``4T-relation'' (and in the
unframed case, the ``1T-'' or ``isolated chord relation''), it is a problem of
some interest to provide a basis for the space of chord diagrams modulo the
4T-relation.
  We construct the basis for the vector space spanned by chord diagrams with n
chords and m distinguishable link components, modulo 4T relations for n less
than or equal to 5.
