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.

翻译：我们研究了在Kendall-τ度量下纠错码的最大基数及其数学结构。我们建立了具有指定最小距离的码的基数平均界，讨论了该界限的紧致性，并刻画了达到该界限的码。这引出了在Kendall-τ度量下t-平衡码族的引入。这些结果基于新颖的论证，为Kendall-τ度量空间的结构提供了新的见解。