On the Dimension of the Hull

The hull Assmus, Jr. and Key, Discrete Math., 83 (1990), pp. 161--187, Assmus, Jr. and Key, Designs and Their Codes, Cambridge University Press, 1992, p. 43 of a linear code is defined to be its intersection with its dual. We give here the number of distinct q-ary linear codes which have a hull of given dimension. We will prove that, asymptotically, the proportion of q-ary codes whose hull has dimension l is a positive constant that depends only on l and q and consequently that the average dimension of the hull is asymptotically a positive constant depending only on q.