Additive one-rank hull codes over finite fields

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 $1\leq n\leq 10$.