The classification of orthogonal arrays OA(2048,14,2,7) and some completely regular codes

TL;DR

This work solves the long-standing problem of classifying binary orthogonal arrays $OA(2048,14,2,7)$ by exploiting a recursive local-code framework, ILP-assisted continuations, and isomorph rejection. The authors partition the search into square and square-free families, enabling a tractable computation that yields $30848$ equivalence classes of $OA(2048,14,2,7)$, with eight square-free instances and $14960$ arising from punctured $1$-perfect codes. They also classify derived arrays $OA(1024,13,2,6)$ and establish the existence of unique almost-OA extensions, along with detailed insights into distance-2 components and derived group divisible designs. The results deepen the connection between orthogonal arrays and completely regular codes, providing a comprehensive catalog and tools for future investigations in coding theory and design theory.

Abstract

We describe the classification of orthogonal arrays OA$(2048,14,2,7)$, or, equivalently, completely regular $\{14;2\}$-codes in the $14$-cube ($30848$ equivalence classes). In particular, we find that there is exactly one almost-OA$(2048,14,2,7{+}1)$, up to equivalence. As derived objects, OA$(1024,13,2,6)$ ($202917$ classes) and completely regular $\{12,2;2,12\}$- and $\{14, 12, 2; 2, 12, 14\}$-codes in the $13$- and $14$-cubes, respectively, are also classified. Keywords: binary orthogonal array, completely regular code, binary 1-perfect code.

Theorems & Definitions (11)

• Lemma 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• Proposition 2.5
• Theorem 3.1: computational
• Theorem 3.2: computational
• Theorem 3.3: computational
• Theorem 4.1