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 p m over F p r which are invariant under AGL ( m e , F p e ) are characterized by a well-known relation ≪ e on the set { 0 , 1 , … , p m - 1 } . From the relation ≪ e , 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 r th 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 , F p 2 ) -invariant extended cyclic codes of length p m over F p r and for the dimensions of such codes. We also enumerate all self-dual AGL ( m 2 , F 2 2 ) -invariant extended cyclic codes of length 2 m over F 2 2 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 .