Weighted-Hamming Metric: Bounds and Codes

TL;DR

This paper introduces bounds and a new code construction for the weighted-Hamming metric, enhancing error correction in applications with variable importance or noise levels across channels.

Contribution

It establishes tighter bounds on error correction under the weighted-Hamming metric and proposes an efficient, flexible code construction based on generalized concatenation.

Findings

Derived direct bounds on error-correction capability

Proposed a generalized concatenation code construction

Demonstrated efficient decoding up to a lower bound

Abstract

The weighted-Hamming metric generalizes the Hamming metric by assigning different weights to blocks of coordinates. It is well-suited for applications such as coding over independent parallel channels, each of which has a different level of importance or noise. From a coding-theoretic perspective, the actual error-correction capability of a code under this metric can exceed half its minimum distance. In this work, we establish direct bounds on this capability, tightening those obtained via minimum-distance arguments. We also propose a flexible code construction based on generalized concatenation and show that these codes can be efficiently decoded up to a lower bound on the error-correction capability.