Abstract
Linear \(\ell \)-intersection pairs of codes serve as a generalization of linear complementary pairs of codes and hulls. The \(\ell \) represents the dimension of the intersection of a given pair of codes over a finite field \(\mathbb {F}_q\). In this paper, we study the linear \(\ell \)-intersection pair of cyclic codes and quasi-cyclic codes over a finite field \(\mathbb {F}_q\). For a given pair of cyclic codes, we derive the value of \(\ell \) in terms of the degrees of the generator polynomials. A construction for an MDS \(\ell \)-intersection of a pair of cyclic codes is presented. Also, a condition for the \(\ell \) intersection to be LCD is derived. In the latter part, we study the \(\ell \)-intersection pair of 1-generator quasi-cyclic codes of index 2. We present a construction for the \(\ell \)-intersection pair of 1-generator quasi-cyclic codes and prove a necessary and sufficient condition for the 1-generator quasi-cyclic code of index 2 to be self-dual. Our result shows that (Esmaeili and Yari in Finite Fields Appl 15(3):375–386, 2009, Theorem 7) is true only for trivial codes. In the end, a construction of 1-EAQEC MDS code is given as an application of our study.
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Acknowledgements
The first author would like to thank IIIT Naya Raipur for the financial support to carry out this work. The second author is supported by the National Board of Higher Mathematics, Department of Atomic Energy, India through project No. 02011/20/2021/ NBHM(R.P)/R &D II/8775.
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Hossain, M.A., Bandi, R. Linear \(\ell \)-intersection pairs of cyclic and quasi-cyclic codes over a finite field \({\mathbb {F}}_q\). J. Appl. Math. Comput. 69, 2901–2917 (2023). https://doi.org/10.1007/s12190-023-01861-z
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DOI: https://doi.org/10.1007/s12190-023-01861-z