MacWilliams identities for poset level weight enumerators of linear codes

TL;DR

This paper introduces new poset level weight enumerators for linear codes over Frobenius rings and derives MacWilliams-type identities that generalize previous results for various metrics.

Contribution

It presents novel poset level weight enumerators and establishes MacWilliams identities for them, extending the applicability to codes over Frobenius rings and generalizing prior identities.

Findings

Derived MacWilliams identities for new poset level weight enumerators.

Generalized previous identities for Hamming, RT, and other metrics.

Applicable to codes over Frobenius rings.

Abstract

Codes over various metrics such as Rosenbloom-Tsfasman (RT), Lee, etc. have been considered. Recently, codes over poset metrics have been studied. Poset metric is a great generalization of many metrics especially the well-known ones such as the RT and the Hamming metrics. Poset metric can be realized on the channels with localized error occurrences. It has been shown that MacWilliams identities are not admissible for codes over poset metrics in general [Kim and Oh, 2005]. Lately, to overcome this problem some further studies on MacWilliams identities over poset metrics has been presented. In this paper, we introduce new poset level weight enumerators of linear codes over Frobenius commutative rings. We derive MacWilliams-type identities for each of the given enumerators which generalize in great deal the previous results discussed in the literature. Most of the weight enumerators in the literature such as Hamming, Rosenbloom-Tsfasman and complete m-spotty weight enumerators follow as corollaries to these identities especially.