Linear Codes Correcting Repeated Bursts Equipped with Homogeneous Distance

The homogeneous weight (metric) is useful in the construction of codes over a ring of integers \(\mathbb {Z}_{p^l}\) ${\mathbb{Z}}_{p^l}$ (p prime and \(l \ge 1\) $l\ge 1$ an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be \(\mathbb {Z}_{4}\) ${\mathbb{Z}}_4$ . This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of n-tuples over \(\mathbb {Z}_{p^l}\) ${\mathbb{Z}}_{p^l}$ . Necessary and sufficient conditions for existence of an (n, k) linear code over \(\mathbb {Z}_{p^l}\) ${\mathbb{Z}}_{p^l}$ correcting the error patterns with respect to the homogeneous weight are derived.