Cyclic codes and minimal strong Gröbner bases over a principal ideal ring

Cyclic and negacyclic codes over the Galois ring GR(p2,m)

“…According to Theorem 4.2 of [16], I admits a set of generators of the form {A(X), pB(X )} such that B(X ) | A(X ) | X N −1 over F p m and we have deg(B(X )) < deg(A(X )), where A(X ) stands for A(X ) mod p. It is easily seen that B(X ) = h(X ) and…”

“…But when p divides N the characterization of the corresponding codes is not that easy. Substantial research work concerning these codes has been developed in [2,4,11,12,16]. Cyclic codes over Z 4 of length 2n with n odd, were studied in [4], and the codes of length 2 n were examined in [2], while codes of arbitrary even length were studied in [11].…”

“…They even considered cyclic codes of arbitrary length over the ring Z p e , but with a cumbersome method leading to a complicated generators. Another form of generators for cyclic codes over a finite chain ring can be found in [6,15,16,18]. These generators are based on the concept of Gröbner basis and seems to be simpler.…”

Dual and self-dual negacyclic codes of even length over Z2a

“…Repeated-root cyclic codes over finite rings have been studied extensively in recent years [1,4,8,19]. The structure of cyclic codes over a finite ring for arbitrary lengths was derived in [14] using Gröbner basis techniques. Meanwhile, attention has also been paid to repeated-root negacyclic codes over finite rings.…”

Dynamical Gröbner bases

“…In the first paragraph of this paper we will extend these notions of Gröbner basis and S-polynomials to Noetherian valuation rings, that is without supposing that the basic ring is integral. This is important for applications especially applications to coding theory as codes over Z/4Z or more generally over Galois rings [5,[13][14][15][16].…”