The size of m -irreducible blocking sets and of the sets of class [0,n1,…,nl][0,n1,…,nl]

We provide an upper bound of the size of an m  -irreducible blocking set in a linear space. This upper bound is a generalization of the Bruen–Thas bound in πqπq and improves it if m>(q2+q-qq)/(qq+1). We prove that in a finite affine plane αqαq of order q  , two blocking sets mutually complementary are both irreducible, if and only if q=4q=4. Moreover, we determine bounds of the size of a set of class [0,n1,…,nl][0,n1,…,nl] in πqπq, ni≡1modd, i=1,…,li=1,…,l, 2⩽d