On Linear Complementary Pair of $n$ D Cyclic Codes

The security parameter for a linear complementary pair (C, D) of codes is defined to be the minimum of the minimum distances d(C) and d(D
^⊥
). Recently, Carlet et al. showed that if C and D are both cyclic or both 2-D cyclic linear complementary pair of codes, and then, C and D
^⊥
are equivalent codes. Hence, the security parameter for cyclic and 2-D cyclic linear complementary pair of codes is simply d(C). We extend this result to nD cyclic linear complementary pair of codes. The proof of Carlet et al. for the 2-D cyclic case is based on the trace representation of the codes, which is technical and nontrivial to generalize. Our proof for the generalization is based on the zero sets of the ideals corresponding to nD cyclic codes.