Repeated-root cyclic and negacyclic codes over a finite chain ring

Abstract

We show that repeated-root cyclic codes over a finite chain ring are in general not principally generated. Repeated-root negacyclic codes are principally generated if the ring is a Galois ring with characteristic a power of 2. For any other finite chain ring they are in general not principally generated. We also prove results on the structure, cardinality and Hamming distance of repeated-root cyclic and negacyclic codes over a finite chain ring.

Some results on cyclic codes over the ring $R_{2,m}$

“…Many of the works in the literature deal with the case wherein R is a finite chain ring [1][2][3][4][5][6][7][8][9][12][13][14][15][16][17][18].…”

On repeated-root multivariable codes over a finite chain ring

“…Proof The result follows directly from the previous theorem and the proof of the result in [17,Theorem 3.4]. …”

“…The particular case where R is the finite field F q was considered in [2, Theorem 1]. The case of cyclic and negacyclic codes was studied in [17,Theorem 3.4]. Finally, the case of multivariable serial codes was studied in [11].…”

Negacyclic self-dual codes over finite chain rings

“…In particular, cyclic and negacyclic codes over finite rings have received much attention (cf. [3,[5][6][7][8]19,22]). Calderbank and Sloane [4] obtained the structure of cyclic codes over Z p m .…”