Bounds for complexity of syndrome decoding for poset metrics

TL;DR

This paper investigates the complexity bounds of syndrome decoding for codes under poset metrics, introducing a decomposition approach and relating it to poset refinements to establish bounds.

Contribution

It introduces a method to decompose codes relative to poset metrics and links poset refinements to decoding complexity bounds.

Findings

Decomposition of codes relative to poset metrics is possible.

Refinement of posets affects the primary decomposition.

Bounds for syndrome decoding complexity are established.

Abstract

In this work we show how to decompose a linear code relatively to any given poset metric. We prove that the complexity of syndrome decoding is determined by a maximal (primary) such decomposition and then show that a refinement of a partial order leads to a refinement of the primary decomposition. Using this and considering already known results about hierarchical posets, we can establish upper and lower bounds for the complexity of syndrome decoding relatively to a poset metric.