  This article explores additive codes with one-rank hull, offering key
insights and constructions. The article introduces a novel approach to finding
one-rank hull codes over finite fields by establishing a connection between
self-orthogonal elements and solutions of quadratic forms. It also 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. Furthermore, the new
quaternary one-rank hull codes are identified over non-symmetric dualities with
better parameters than symmetric ones.
