The Existence of MacWilliams-Type Identities for the Lee, Homogeneous and Subfield Metric

Abstract

Famous results state that the classical MacWilliams identities fail for the Lee metric, the homogeneous metric and for the subfield metric, apart from some trivial cases. In this paper we change the classical idea of enumerating the codewords of the same weight and choose a finer way of partitioning the code that still contains all the information of the weight enumerator of the code. The considered decomposition allows for MacWilliams-type identities which hold for any additive weight over a finite chain ring. For the specific cases of the homogeneous and the subfield metric we then define a coarser partition for which the MacWilliams-type identities still hold. This result shows that one can, in fact, relate the code and the dual code in terms of their weights, even for these metrics. Finally, we derive Linear Programming bounds stemming from the MacWilliams-type identities presented.