$t$-Balanced Codes with the Kendall-$\tau$ Metric

TL;DR

This paper explores the structure and limits of error-correcting codes under the Kendall-$\tau$ metric, introducing $t$-balanced codes and providing bounds on their size and properties.

Contribution

It introduces the concept of $t$-balanced codes in the Kendall-$\tau$ metric and characterizes their maximum size and structure, advancing understanding of Kendall-$\tau$ error correction.

Findings

Established an averaging bound for code cardinality with minimum distance

Characterized codes that attain the bound

Introduced and analyzed $t$-balanced codes in the Kendall-$\tau$ metric

Abstract

We investigate the maximum cardinality and the mathematical structure of error-correcting codes endowed with the Kendall-$\tau$ metric. We establish an averaging bound for the cardinality of a code with prescribed minimum distance, discuss its sharpness, and characterize codes attaining it. This leads to introducing the family of $t$-balanced codes in the Kendall-$\tau$ metric. The results are based on novel arguments that shed new light on the structure of the Kendall-$\tau$ metric space.