Structure of codes with poset metric

Coding Theory has been a very important field of study in terms of transmission and storage of messages. While transmitting the messages through a noisy channel, there may be many errors which the messages incur. In this situation, the receiver gets the incorrect message. In order to detect and correct the error, the receiver applies the Theory of Error Correcting Codes.During the encoding of the messages, weight and distance functions are defined according to the type of errors. By means of these functions, decoding is done depending on the distance between the original sent message and the received erroneous word. The distance function being a metric, facilitates detecting and correcting the errors.One of the oldest and the most applied distance functions is the Hamming distance. However, several distance functions are defined since the type of errors may be different.The MacWilliams identity, which gives the relation between the weight enumerators of the linear codes and their duals was first proved for the Hamming metric. This identity has a significant role in Algebraic Coding Theory. Hence, some new MacWilliams type identities have been introduced for new metrics since then.Recently, the poset metric has been a reference for many works in Coding Theory since as a special case it gives the well known Hamming and Rosenbloom-Tsfasmann metrics.In this work, poset metric has been studied, a new weight enumerator is defined which is called P- complete weight enumerator and uses only level details of the poset comprehensively instead of weight enumerator which is defined for hierarchical posets. According to this weight enumerator, MacWilliams type identity has been proved. We obtain the well known weight enumerators of Hamming and RT- metrics, as a special case of the newly defined weight enumerator.Key words: Poset, Partially Ordered Set, code, weight, MacWilliams Identitiy, Linear Codes, Weight Enumerators