This paper investigates the hull codes of free linear codes 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$. Initially, we examine the residue and torsion codes of various hulls of $E$-linear codes and obtain an explicit form of the generator matrix of the hull of a free $E$-linear code. Then, we propose four build-up construction methods to construct codes with a larger length and hull-rank from codes with a smaller length and hull-rank. Some illustrative examples are also given to support our build-up construction methods. Subsequently, we study the permutation equivalence of two free $E$-linear codes and discuss the hull-variation problem. As an application, we classify optimal free $E$-linear codes for lengths up to $8$.

翻译：本文研究了非幺环$E=\langle κ,τ\mid 2κ=2τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ\rangle$上自由线性码的壳码。首先，我们考察了$E$-线性码各类壳的剩余码与挠码，并得到了自由$E$-线性码壳的生成矩阵的显式形式。接着，我们提出了四种递升构造方法，用于从具有较小长度和壳秩的码构造具有更大长度和壳秩的码。文中也给出了一些说明性示例以支持我们的递升构造方法。随后，我们研究了两个自由$E$-线性码的置换等价性，并讨论了壳变化问题。作为应用，我们对长度不超过$8$的最优自由$E$-线性码进行了分类。