Additive one-rank hull codes over finite fields

Abstract

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 Fq, 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 d1[n,k]pe,M. The value of d1[n,k]pe,M for k=1,2 and n≥2 with respect to any duality M over any finite field Fpe is determined. Furthermore, the new quaternary one-rank hull codes are identified over non-symmetric dualities with better parameters than symmetric ones.