$t$-Balanced Codes with the Kendall-$τ$ Metric
Benjamin Jany, Alberto Ravagnani
TL;DR
This work studies error-correcting codes under the Kendall-$\tau$ distance on the permutation group $\mathbb{S}_n$, aiming to bound the maximum code size for a given length and minimum distance. It introduces an averaging bound derived via puncturing and an averaging argument, defines $t$-balanced codes that meet the bound with equality, and proves such codes exist only for $t=2$, where they are exactly translates of the even-permutation subgroup $A_n$ and hence Kendall-$\tau$-MDS. The results illuminate the structure of Kendall-$\tau$ spaces, connect to sphere-packing and code-anticode bounds, and establish a complete classification for $2$-balanced codes while outlining key open problems in full MDS characterization and optimal anticodes for general parameters. These findings have potential implications for rank modulation schemes and related permutation-based coding applications by providing tight bounds and explicit constructions.
Abstract
We investigate the maximum cardinality and the mathematical structure of error-correcting codes endowed with the Kendall-$τ$ 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-$τ$ metric. The results are based on novel arguments that shed new light on the structure of the Kendall-$τ$ metric space.