On the Dimension of the Hull

The hull AK90],,AK92, p. 43] of a linear code is deened 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 only depends on l and q, and consequently that the average dimension of the hull is asymptotically a positive constant dependent of q. Sur la dimension de l'intersection d'un code et de son orthogonal R esum e : Nous donnons ici le nombre de codes lin eaires de longueur n et de dimension k sur GF(q) ayant une intersection de dimension donn ee avec leur orthogonal. Nous montrons que la dimension moyenne de cette intersection tend vers une constante strictement positive, d ependant de q, lorsque n et k tendent vers l'innni.