Intersection of linear and multi-twisted codes with applications

In this paper, we construct 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 generator matrix for the largest reversible subcode of any linear code. We then focus on the comprehensive class of multi-twisted (MT) codes, which are more effectively represented using generator polynomial matrices (GPMs). We prove that the reversed code of an MT code remains MT and derive 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 admits an MT structure, we propose the corresponding shift constants. We also establish a GPM formula for the intersection of a pair of MT codes with identical shift constants. This result yields a GPM formula for the intersection of an MT code and the Galois dual of another MT code. Finally, we examine the necessary and sufficient conditions for an MT code to be Galois self-orthogonal, Galois dual-containing, Galois linear complementary dual (LCD), or reversible.