Additive one-rank hull codes over finite fields

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 $F_q$, 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 $n\ge 2$ with respect to any duality M over any finite field $F_{p^e}$ is determined. Furthermore, the new quaternary one-rank hull codes are identified over non-symmetric dualities with better parameters than symmetric ones.