In 1998, E. Couselo, S. González, V. T. Markov, and A. A. Nechaev introduced the notions of recursive codes and recursively differentiable quasigroups. They conjectured that recursive MDS codes of dimension $2$ and length $4$ exist over every finite alphabet of size $q \not\in \{2, 6\}$, and verified this conjecture in all cases except $q \in \{14, 18, 26, 42\}$. In 2008, V. T. Markov, A. A. Nechaev, S. S. Skazhenik, and E. O. Tveritinov resolved the case $q=42$ by providing an explicit construction. The present paper settles the outstanding case $q=26$. The construction rests upon methods for producing recursively differentiable quasigroups and recursive MDS codes via perfect cyclic Mendelsohn designs. Moreover, we sharpen several known bounds concerning the existence of recursively $n$-differentiable quasigroups of small orders.

翻译：1998年，E. Couselo、S. González、V. T. Markov 和 A. A. Nechaev 引入了递归码与递归可微拟群的概念。他们猜想，对于任意大小为 $q \not\in \{2, 6\}$ 的有限字母表，存在维度为 $2$、长度为 $4$ 的递归 MDS 码，并在除 $q \in \{14, 18, 26, 42\}$ 外的所有情形下验证了这一猜想。2008年，V. T. Markov、A. A. Nechaev、S. S. Skazhenik 和 E. O. Tveritinov 通过提供显式构造解决了 $q=42$ 的情况。本文解决了剩余未解的 $q=26$ 情形。该构造基于通过完美循环 Mendelsohn 设计生成递归可微拟群与递归 MDS 码的方法。此外，我们改进了若干关于小阶递归 $n$-可微拟群存在性的已知界值。