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

04/05/2023

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by   Zohreh Aliabadi, et al.

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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 zero hull dimension is called a linear complementary dual (LCD) code. A pair (C, D) of linear codes of length n over 𝔽_q is called a linear complementary pair (LCP) of codes if C⊕ D=𝔽_q^n. In this paper, we give a characterization of LCD and LCP of cyclic codes of length q^m-1, m ≥ 1, over the finite field 𝔽_q in terms of their basic dual zeros and their trace representations. We also formulate the hull dimension of a cyclic code of arbitrary length over 𝔽_q with respect to its basic dual zero. Moreover, we provide a general formula for the dimension of the intersection of two cyclic codes of arbitrary length over 𝔽_q based on their basic dual zeros.