Embedding linear codes over Z4 into self-orthogonal codes
2026-06-08 • Information Theory
Information Theory
AI summaryⓘ
The authors study how to embed linear codes over the ring Z4 into self-orthogonal codes, which are important for error correction. They find tight bounds and exact lengths for these embeddings under certain conditions, including for the special case of quaternary Preparata codes. To support this, they fully classify similar embeddings for binary codes. Additionally, they create an algorithm to build all shortest embeddings when certain length conditions are met, which helped them discover new codes with better error-correcting properties than some known examples.
linear codesZ4self-orthogonal codesembedding lengthPreparata codesbinary linear codesdoubly even codesLee distanceerror-correcting codescode construction algorithms
Authors
Junmin An, Jon-Lark Kim, San Ling
Abstract
The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.