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.

翻译：本文研究了在非单位环$E=\langle κ,τ\mid 2κ=2τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ\rangle$上的辛包络。我们首先确定了左、右及双侧辛包络的剩余码与挠码，并刻画了自由$E$-线性码的双侧辛包络的生成矩阵。随后，我们探讨了两个自由$E$-线性码之和的辛包络。接着，我们提出了两种构造方法，可将具有较小长度和辛包络秩的自由$E$-线性码扩展为具有较大长度和辛包络秩的码。此外，针对自由$E$-线性码，我们讨论了置换等价性并研究了辛包络变化问题。本研究的一个应用是通过分类较短长度下的自由$E$-线性最优码来体现。