MDS and I-Perfect Codes in Pomset Metric

Abstract

Recently pomset metric was introduced to accommodate Lee metric, thereby generalizing poset metrics for codes over $\mathbb {Z}_{{m}}$ . This work mainly studies results on Maximum distance separable (MDS) pomset codes and perfect codes over $\mathbb {Z}_{{m}}$ . If I is an ideal in a poset, the concept of I -perfect codes that was introduced to study MDS poset codes is now extended to the study of MDS pomset codes as well. But, unlike the ideals in posets, here the ideals in pomsets are considered as those with full count and those with partial count. In fact, the I -balls are no more linear when the ideal I is with partial count. This makes the investigation of the relationship of I -perfect codes with MDS pomset codes little different from that with MDS poset codes. We thoroughly study the I -balls where I is with partial count. Singleton type bound is established for codes with pomset metric and the connection of MDS codes with I -perfect codes is investigated upon. Then, we prove the duality theorem for codes with chain pomset. Finally, we determine the weight distribution of MDS pomset codes when the pomset is a chain.