Generator polynomial matrices of the Galois hulls of multi-twisted codes
In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field 𝔽_p^e of characteristic p. Let 𝐆 be a generator polynomial matrix (GPM) of a MT code 𝒞. For any 0≤κ<e, the κ-Galois hull of 𝒞, denoted by h_κ(𝒞), is the intersection of 𝒞 with its κ-Galois dual. The main result in this paper is that a GPM for h_κ(𝒞) has been obtained from 𝐆. We start by associating a linear code 𝒬_𝐆 with 𝐆. We show that 𝒬_𝐆 is quasi-cyclic. In addition, we prove that the dimension of h_κ(𝒞) is the difference between the dimension of 𝒞 and that of 𝒬_𝐆. Thus the determinantal divisors are used to derive a formula for the dimension of h_κ(𝒞). Finally, we deduce a GPM formula for h_κ(𝒞). In particular, we handle the cases of κ-Galois self-orthogonal and linear complementary dual MT codes; we establish equivalent conditions that characterize these cases. Equivalent results can be deduced immediately for the classes of cyclic, constacyclic, quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they are all special cases of MT codes. Some numerical examples, containing optimal and maximum distance separable codes, are used to illustrate the theoretical results.
READ FULL TEXT