This article explores additive codes with one-rank hull, offering key insights and constructions. It gives a characterization of the hull of an additive code $C$ in terms of its generator matrix and establishes a connection between self-orthogonal elements and solutions of quadratic forms. Using self-orthogonal elements, the existence of a one-rank hull code is demonstrated. The article provides a precise count of self-orthogonal elements for any duality over the finite field $\mathbb{F}_q$, particularly odd primes. Additionally, construction methods for small-rank hull codes are introduced. The highest possible minimum distance among additive one-rank hull codes is denoted by $d_1[n,k]_{p^e,M}$. The value of $d_1[n,k]_{p^e,M}$ for $k=1,2$ and $n\geq 2$ with respect to any duality $M$ over any finite field $\mathbb{F}_{p^e}$ is determined. Also, the highest possible minimum distance for Quaternary one-rank hull code is determined over non-symmetric dualities for length $1\leq n\leq 10$.

翻译：本文研究了具有单秩壳的加性码，提供了关键见解与构造方法。通过生成矩阵刻画了加性码$C$的壳，并建立了自正交元与二次型解之间的联系。利用自正交元，证明了单秩壳码的存在性。本文给出了有限域$\mathbb{F}_q$上任意对偶性下自正交元的精确计数，特别针对奇素数情形。此外，引入了小秩壳码的构造方法。用$d_1[n,k]_{p^e,M}$表示加性单秩壳码所能达到的最高最小距离。对于任意有限域$\mathbb{F}_{p^e}$上的任意对偶性$M$，确定了$k=1,2$且$n\geq 2$时$d_1[n,k]_{p^e,M}$的值。同时，在长度$1\leq n\leq 10$的非对称对偶性下，确定了四元单秩壳码所能达到的最高最小距离。