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

Petr Klimov

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.

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 and length exist over every finite alphabet of size , and verified this conjecture in all cases except . In 2008, V. T. Markov, A. A. Nechaev, S. S. Skazhenik, and E. O. Tveritinov resolved the case by providing an explicit construction. The present paper settles the outstanding case . 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 -differentiable quasigroups of small orders.

Paper Structure

This paper contains 5 sections, 16 theorems, 29 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Cyclic Mendelsohn Designs
  3. Construction of Recursively Differentiable Quasigroups via Combinatorial Designs
  4. Construction of a Recursively Differentiable Quasigroup of Order 26
  5. A Recursively Differentiable Quasigroup of Order 26

Key Result

Theorem 1.1

MarkovBase A complete $2$-recursive code of length $n \ge 3$, specified by a function $f$, is MDS if and only if the corresponding groupoid $(\Omega, f)$ is a recursively $(n -3)$-differentiable quasigroup. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Block construction scheme.

Theorems & Definitions (37)

  • Theorem 1.1
  • Conjecture 1.2: Couselo-González-Markov-Nechaev
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 3.1
  • proof
  • ...and 27 more