Weights which respect support and NN-decoding

TL;DR

This paper introduces a new family of support-respecting metrics over finite fields, explores their properties, and characterizes their isometries and conditions for MacWilliams identities.

Contribution

It presents the labeled-poset block metrics, filling a gap in support-respecting metrics, and analyzes their isometries and MacWilliams identity conditions.

Findings

Introduced labeled-poset block metrics respecting support.

Characterized the group of linear isometries for these metrics.

Determined conditions for the existence of MacWilliams identities.

Abstract

In this work we explore a family of metrics over finite fields which respect the support of vectors. We show how these metrics can be obtained from the edge-weighted Hamming cube and, based on this representation we give a description of a group of linear isometries (with respect to the metric). Next we introduce the concept of conditional sum of metrics and determine what conditions determine a metric respecting support, out of two such given metrics. Finally we introduce the labeled-poset block metrics, a new family of metrics which respects support of vector, filling a gap existing in the known such metrics. For this family we give a full description of the group of linear isometries and determine necessary and sufficient conditions for the existence of a MacWilliams identity.