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

Abstract: 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.

• The paper explicitly constructs a recursively differentiable quasigroup, confirming the existence of a recursive [4,2,3]₍₂₆₎-code using cyclic Mendelsohn designs.
• It employs perfect cyclic Mendelsohn designs and recursive derivatives to establish improved lower bounds for MDS codes over finite alphabets.
• The approach bridges algebraic coding theory with combinatorial design, offering advancements in cryptography and redundancy systems.

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

Overview and Historical Context

The paper addresses the outstanding case in the problem of constructing recursively differentiable quasigroups and recursive MDS codes of dimension $2$ and length $4$ over a finite alphabet of size $26$. Originating from the 1998 conjecture of Couselo–González–Markov–Nechaev, which hypothesized the existence of recursive $[4,2,3]_q$-codes for every $q \notin \{2,6\}$, previous constructions had left the cases $q \in \{14, 18, 26, 42\}$ unresolved. The case $q=42$ was settled explicitly in 2008; this work explicitly constructs the case $q=26$, thereby affirmatively resolving Conjecture \ref{hyp} for all but two trivial exceptions.

Foundations: Quasigroups, Recursive Codes, and the Singleton Bound

The investigation is based on the combinatorial interplay between codes, Latin squares, and quasigroups. The central objects are $[n,k,d]_q$-codes and their maximal distance separable (MDS) instances, which saturate the Singleton bound $2$0.

A recursively differentiable quasigroup is defined via recursive derivatives: starting with a binary operation $2$1, the sequence of recursive derivatives $2$2 is defined by iteratively combining previous operations. A quasigroup is recursively $2$3-differentiable if all derivatives up to order $2$4 are quasigroups.

A central equivalence (Theorem 1) is established: the existence of a complete $2$5-recursive MDS code of length $2$6 is equivalent to the existence of a recursively $2$7-differentiable quasigroup.

Construction Method: Cyclic Mendelsohn Designs

The main construction utilizes perfect cyclic Mendelsohn designs (PMDs), combinatorial objects that guarantee the uniqueness and cyclic orderings necessary for the construction of quasigroups with high recursive differentiability. The directed standard construction maps cyclically ordered blocks directly to binary operations on the corresponding point set and allows for rigorous control over the differentiability indices.

A series of theorems formalize the relationship between the cyclic decomposition of the corresponding right quasigroup, the properties of the PMD, and the recursive derivatives, culminating in a sharp upper bound for the achievable degree of recursive differentiability in Theorem \ref{e}: $2$8-PMDs yield recursively $2$9-differentiable quasigroups, which is maximal.

Explicit Construction for Order 26

The explicit construction employs a $4$0-PMD, drawing upon the combinatorial framework developed by Bennett and others. The alphabet is constructed as $4$1. The collection of blocks combines cyclic permutations with the addition of blocks among the $4$2-indices, guaranteeing $4$3-apartness for all ordered pairs of distinct elements. Application of the directed standard construction yields a recursively $4$4-differentiable idempotent quasigroup structure.

The paper provides full Cayley tables for the quasigroup and its first two recursive derivatives, explicitly verifying the recursive differentiability up to order $4$5. This construction confirms the existence of a recursive $4$6-code, resolving the final nontrivial case of a problem open since 1998.

Numerical Implications and Improved Bounds

The approach yields new, strengthened lower bounds for the maximal length $4$7 achievable by recursive MDS codes for various $4$8 congruence classes, with precise exceptions tabulated (see Theorem \ref{hu}). For $4$9, $26$0 is established.

These results improve known lower bounds on the degree of recursive differentiability for small $26$1; a detailed table enumerates the previously best-known and improved values up to $26$2, establishing the utility of the combinatorial construction methodology.

Theoretical and Practical Implications

The explicit link between combinatorial design theory and recursive algebraic code properties solidifies the role of Mendelsohn designs in code and quasigroup construction. The methodology demonstrates that for sufficiently large and appropriately structured alphabets, recursive MDS codes exist up to the strict limitations imposed by the order of the underlying combinatorial design.

Practically, these codes are of interest for structured redundancy in coding theory, cryptographic primitives, and combinatorial algebra, where quasigroup structures underpin functional completeness and recursive evaluation.

Speculation on Future Directions

The techniques presented suggest that further progress may be made by extending the Mendelsohn design framework to more general block sizes or relaxing perfectness conditions, potentially addressing open existence and construction problems for recursive codes of even higher dimension and alphabet size. The relationship between the explicit cycle constructions and recursive differentiability metrics will likely provide leverage for developing new classes of codes and algebraic structures of interest in combinatorics and theoretical computer science.

Conclusion

This work definitively resolves the existence of a recursively differentiable quasigroup—and hence a recursive $26$3-MDS code—over an alphabet of size $26$4. The methodology systematically leverages perfect cyclic Mendelsohn designs to guarantee recursive differentiability, yielding new and improved bounds for the maximal length of recursive MDS codes across small alphabets. The results deepen the interaction between combinatorial design theory and recursive algebraic structures, with tangible implications for algebraic coding and combinatorics.

Reference: "On the Construction of Recursively Differentiable Quasigroups and an Example of a Recursive $26$5-Code" (2604.01105)