Enumeration of certain affine invariant extended cyclic codes

摘要

Let p be a prime and let r, e, m be positive integers such that r vertical bar e and elm. 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 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, F-p(2))-invariant extended cyclic codes of length p(m) over Fpr 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. (c) 2004 Elsevier Inc. All rights reserved.