Symplectic Hulls over a Non-Unital Ring
Anup Kushwaha, Om Prakash
TL;DR
The paper addresses symplectic hulls of $E$-linear codes over the non-unital ring $E$ with generators $\,\kappa,\tau$ (where $2\kappa=2\tau=0$, $\kappa^2=\kappa$, $\tau^2=\tau$, $\kappa\tau=\kappa$, $\tau\kappa=\tau$). It develops a framework for left, right, and two-sided hulls, derives residue and torsion codes, and provides a generator-matrix description for $SHull(C)$ of free $E$-linear codes, including the rank relation $\mathrm{rank}(SHull(C))=\mathrm{rank}(SHull(C_{Res}))$. Two build-up constructions enlarge length and hull-rank while preserving freeness of hulls, and the paper analyzes permutation-invariance of hull-ranks, proving invariance under symplectic-preserving permutations and presenting counterexamples otherwise. It further classifies free $E$-linear optimal codes for small lengths with explicit constructions. Overall, the work extends hull-based coding theory to non-unital rings and opens directions for studying symplectic structures over other rings in Fine’s classification.
Abstract
This paper presents the study of the symplectic hulls over a non-unital ring $ E= \langle κ,τ\mid 2 κ=2 τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ\rangle$. We first identify the residue and torsion codes of the left, right, and two-sided symplectic hulls, and characterize the generator matrix of the two-sided symplectic hull of a free $E$-linear code. Then, we explore the symplectic hull of the sum of two free $E$-linear codes. Subsequently, we provide two build-up techniques that extend a free $E$-linear code of smaller length and symplectic hull-rank to one of larger length and symplectic hull-rank. Further, for free $E$-linear codes, we discuss the permutation equivalence and investigate the symplectic hull-variation problem. An application of this study is given by classifying the free $E$-linear optimal codes for smaller lengths.