In this paper, we derive a formula for constructing a generator matrix for the intersection of any pair of linear codes over a finite field. Consequently, we establish a condition under which a linear code has a trivial intersection with another linear code (or its Galois dual). Furthermore, we provide a condition for reversibility and propose a generator matrix formula for the largest reversible subcode of any linear code. We then focus on the comprehensive class of multi-twisted (MT) codes, which are naturally and more effectively represented using generator polynomial matrices (GPMs). We prove that the reversed code of an MT code remains MT and derive an explicit formula for its GPM. Additionally, we examine the intersection of a pair of MT codes, possibly with different shift constants, and demonstrate that this intersection is not necessarily MT. However, when the intersection has an MT structure, we determine the corresponding shift constants. We also establish a GPM formula for the intersection of a pair of MT codes with the same shift constants. This result enables us to derive a GPM formula for the intersection of an MT code and the Galois dual of another MT code. Finally, we examine conditions for various properties on MT codes. Perhaps most importantly, the necessary and sufficient conditions for an MT code to be Galois self-orthogonal, Galois dual-containing, Galois linear complementary dual (LCD), or reversible.

翻译：本文推导了在有限域上构造任意一对线性码交集生成矩阵的公式。由此，我们建立了线性码与另一线性码（或其伽罗瓦对偶码）具有平凡交集的条件。进一步，我们给出了可逆性的判定条件，并提出了任意线性码的最大可逆子码的生成矩阵公式。随后，我们聚焦于多扭码这一广泛类别，该类码自然且更有效地通过生成多项式矩阵进行表示。我们证明了多扭码的逆码仍为多扭码，并推导了其生成多项式矩阵的显式公式。此外，我们研究了具有可能不同移位常数的一对多扭码的交集，并证明该交集未必保持多扭结构。然而，当交集具有多扭结构时，我们确定了相应的移位常数。我们还建立了具有相同移位常数的一对多扭码交集的生成多项式矩阵公式。这一结果使我们能够推导出多扭码与另一多扭码的伽罗瓦对偶码交集的生成多项式矩阵公式。最后，我们考察了多扭码多种性质的判定条件。其中最为重要的是，给出了多扭码成为伽罗瓦自正交码、伽罗瓦对偶包含码、伽罗瓦线性互补对偶码或可逆码的充分必要条件。