Intersection of linear and multi-twisted codes with applications

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.