Linear Complementary Equi-Dual Codes

We call a linear code C with length n over a field F, a linear complementary equi-dual code, when there exists a linear code D over F such that D is permutation equivalent to C^⊥ and $(C,D)$ is a linear complementary pair of codes, that is, C+ D=Fⁿ and C∩ D=0. We first state a necessary condition on a code C to be linear complementary equi-dual. Then, we conjecture that this necessary condition is also sufficient and present several statements which support this conjecture.