Self-orthogonal codes are an important subclass of linear codes which have nice applications in quantum codes and lattices. It is known that a binary linear code is self-orthogonal if its every codeword has weight divisible by four, and a ternary linear code is self-orthogonal if and only if its every codeword has weight divisible by three. It remains open for a long time to establish the relationship between the self-orthogonality of a general $q$-ary linear code and the divisibility of its weights, where $q=p^m$ for a prime $p$. In this paper, we mainly prove that any $p$-divisible code containing the all-1 vector over the finite field $\mathbb{F}_q$ is self-orthogonal for odd prime $p$, which solves this open problem under certain conditions. Thanks to this result, we characterize that any projective two-weight code containing the all-1 codeword over $\mathbb{F}_q$ is self-orthogonal. Furthermore, by the extending and augmentation techniques, we construct six new families of self-orthogonal divisible codes from known cyclic codes. Finally, we construct two more families of self-orthogonal divisible codes with locality 2 which have nice application in distributed storage systems.

翻译：自正交码是一类重要的线性码子类，在量子码和格中具有良好应用。已知二进制线性码的每个码字重量可被四整除时即为自正交码，而三元线性码自正交当且仅当其每个码字重量可被三整除。对于一般的$q$元线性码（其中$q=p^m$，$p$为素数），其自正交性与重量整除性之间的关系长期悬而未决。本文主要证明：在奇素数$p$条件下，有限域$\mathbb{F}_q$上包含全1向量的任意$p$-可除码均为自正交码，从而在特定条件下解决了这一开放问题。基于该结果，我们证明了$\mathbb{F}_q$上包含全1码字的任意射影二重量码必为自正交码。进一步，通过扩展与增广技术，我们从已知循环码构造了六个新的自正交可除码族。最后，我们构造了两个在分布式存储系统中具有良好应用前景的局部性为2的自正交可除码族。