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(me,Fpe) are characterized by a well-known relation ≪e on the set {0,1,…,pm-1}. From the relation ≪e, we derive a partial order ≺ in U={0,1,…,me(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(m2,Fp2)-invariant extended cyclic codes of length pm over Fpr and for the dimensions of such codes. We also enumerate all self-dual AGL(m2,F22)-invariant extended cyclic codes of length 2m over F22 where m2 is odd; the restrictions on the parameters are necessary conditions for the existence of self-dual affine invariant extended cyclic codes with e=2.