On LCD codes over \(\mathbb {Z}_4\)

Linear Complementary Dual (LCD) code is a linear code with a trivial intersection with its dual. In this paper, we prove that non-free LCD codes do not exist over \(\mathbb {Z}_4\) ${\mathbb{Z}}_4$ and obtain a necessary and sufficient condition for the existence of LCD codes over \(\mathbb {Z}_4\) ${\mathbb{Z}}_4$ . Later, we investigate free LCD cyclic codes of odd lengths over \(\mathbb {Z}_4\) ${\mathbb{Z}}_4$ and find a relation between free LCD cyclic codes of odd lengths and reversible codes of odd lengths over \(\mathbb {Z}_4\) ${\mathbb{Z}}_4$ . We prove that free LCD cyclic codes of length \(2^k\) $2^k$ , a positive integer k, do not exist, and also establish a condition for the existence of free LCD cyclic codes of lengths \(2^km\) $2^{k}m$ for a positive integer m with \(\gcd (2,m)=1\) $gcd(2,m)=1$ . Finally, we present some good parameters of LCD codes that we obtained.