A note on the hull and linear complementary pair of cyclic codes

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Official URL: https://dx.doi.org/10.55730/1300-0098.3545

The Euclidean hull of a linear code C is defined as C ∩ C⊥, where C⊥ denotes the dual of C under the Euclidean inner product. A linear code with the trivial hull is called a linear complementary dual (LCD) code. A pair (C, D) of linear codes of length n over the finite field \BbbFq is called a linear complementary pair (LCP) of codes if C ⊕ D = \BbbFnq. More generally, a pair (C, D) of linear codes of the same length over \BbbFq is called a linear ℓ-intersection pair of codes if C ∩ D has dimension ℓ as a vector space over \BbbFq. In this paper, we give characterizations of LCD, LCP of cyclic codes and one-dimensional hull cyclic codes of length qm − 1, m ≥ 1, over \BbbFq in terms of their basic dual zero sets and their trace representations. We also formulate the hull dimension of a cyclic code of arbitrary length over \BbbFq with respect to its basic dual zero set. Moreover, we provide a general formula for the dimension ℓ of the intersection of two cyclic codes of arbitrary length over \BbbFq based on their basic dual zero sets.