On the Construction of Recursively Differentiable Quasigroups and an Example of a Recursive $[4,2,3]_{26}$-Code

TL;DR

This paper resolves the open case of recursive MDS codes over a 26-element alphabet by constructing recursively differentiable quasigroups using perfect cyclic Mendelsohn designs.

Contribution

It provides the first explicit construction of recursive MDS codes over a 26-element alphabet, completing the conjecture for all finite sizes except 2 and 6.

Findings

Constructed recursively differentiable quasigroups for q=26.

Resolved the existence of recursive MDS codes over a 26-element alphabet.

Sharpened bounds on small-order recursively n-differentiable quasigroups.

Abstract

In 1998, E. Couselo, S. Gonz\'alez, 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.