Enumeration of certain affine invariant extended cyclic codes

Let p be a prime and let r, e, m be positive integers such that r|e and e|m. Extended cyclic codes of length pm over Fpr which are invariant under AGL(m/e, Fpe) are characterized by a well-known relation le on the set {0, 1,...,pm - 1}. From the relation le, we derive a partial order ≺ in u = {0, 1,...,m/e(p - 1)}e defined by an e-dimensional simplicial cone. We show that the aforementioned extended cyclic codes can be enumerated by the ideals of (u, ≺) which are invariant under the rth power of a circulant permutation matrix. When e = 2, we enumerate all such invariant ideals by describing their boundaries. Explicit formulas are obtained for the total number of AGL(m/2, Fp2)- invariant extended cyclic codes of length pm over Fpr and for the dimensions of such codes. We also enumerate all self-dual AGL(m/2, F22)-invariant extended cyclic codes of length 2m over F22 where m/2 is odd; the restrictions on the parameters are necessary conditions for the existence of self-dual affine invariant extended cyclic codes with e = 2.