Generator polynomial matrices of the Galois hulls of multi-twisted codes
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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.
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