Additive one-rank hull codes over finite fields
Astha Agrawal, R. K. Sharma
TL;DR
This work extends hull-based code analysis to additive codes over finite fields under arbitrary dualities, focusing on one-rank hull codes. It establishes a precise link between hull structure and a quadratic form associated with the duality, enabling exact counts of self-orthogonal elements for odd characteristic dualities and a rank-based criterion for one-rank hulls. The authors develop construction techniques that produce small-rank hull codes from self-orthogonal or ACD codes and derive the maximal possible distances d1[n,k] in key cases, including new near-optimal quaternary codes under non-symmetric dualities. The results broaden the design space for additive codes with small hulls, with potential applications to quantum error correction and related algorithmic contexts.
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 $\mathbb{F}_q$, 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.