$t$-Balanced Codes with the Kendall-$τ$ Metric

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.