A note on cyclic codes over GR(p 2, m) of length p k

Abstract

The number of self-dual cyclic codes of length p k over GR( p 2 , m) is determined by the nullity of a certain matrix M( p k , i 1 ). With the aid of Genocchi numbers, we determine the nullity of M( p k , i 1 ) and hence determine completely the number of such codes.

A note on cyclic codes over GR(p2,m) of length pk

“…The proof proceeds similar steps as those considered in the proof of Theorem 4.2 in [6]. However, in that proof, the authors use the equality…”

“…In this section we review the results presented in Section 3 of [6]. In the rest of this note, S stands for the ring GR(p 2 , m)[u]/ u p k − 1 and I denotes the set of ideals in S. To any element C of I, two ideals of F p m [u]/ u p k − 1 are associated.…”

“…The numbers R(C ) and T (C) will be referred to as the residue degree and the torsion degree of C , respectively. [6] indicates that the map φ : A → A defined by C → Ann(C ) is a bijection. Assume that τ m is the well-known Teichmuller set of coset representatives of GR(p e , m) modulo p and define the set…”

“…Now Theorem 4.1 in [6] indicates that C ⊥ = Ann(C ). Assume that δ(p) = 0 if p = 2 and δ(p) = 1 if p is an odd prime.…”

“…Now we determine self-dual ideals in I for p = 3 and small values of k. The case p = 2 has been investigated in [6,Corollary 4.5]. …”