Euclidean and Hermitian Self-Orthogonal Codes from Divisible Codes
Ziling Heng, Chenkun Xie
原文摘要
Self-orthogonality and divisibility are two fundamental properties of linear codes, and the relationship between them has long been of interest. Huffman and Pless (2003) showed that every binary \(4\)-divisible code is Euclidean self-orthogonal, that a ternary code is Euclidean self-orthogonal if and only if it is \(3\)-divisible, and that a quaternary code is Hermitian self-orthogonal if and only if it is \(2\)-divisible. The general case remained open for a long time. In 2024, Li and Heng settled the odd-characteristic case, proving that every \(p\)-divisible code over \(\F_{p^m}\), with \(p\) odd, that contains the all-one vector is Euclidean self-orthogonal. In this paper we treat the even-characteristic case: we prove that every \(4\)-divisible code over \(\F_{2^m}\) containing the all-one vector is Euclidean self-orthogonal. We also prove that, for every prime \(p\), every \(p\)-divisible linear code over \(\F_{q^2}\), where \(q=p^m\), that contains the all-one vector is Hermitian self-orthogonal. These criteria make the construction of self-orthogonal codes from divisible codes simple and efficient. As applications, we recover the known Euclidean and Hermitian self-orthogonality of generalized Reed--Muller codes and obtain several infinite families of new Euclidean and Hermitian self-orthogonal codes, which give rise to infinite families of quantum codes with excellent parameters.
cs.IT