# The condition for a cyclic code over Z4 of odd length to have a complementary dual

## TL;DR

This paper establishes a precise algebraic condition for cyclic codes over Z4 of odd length to possess a complementary dual, linking the property to the self-reciprocal nature of their generator polynomials.

## Contribution

It provides a necessary and sufficient criterion for cyclic LCD codes over Z4 of odd length based on the self-reciprocal property of the generator polynomial.

## Key findings

- Cyclic LCD codes over Z4 of odd length are generated by self-reciprocal polynomials.
- The paper characterizes LCD codes in terms of polynomial properties in Z4[X].
- A clear algebraic condition for the existence of LCD cyclic codes over Z4 is established.

## Abstract

We show that a necessary and sufficient condition for a cyclic code C over Z4 of odd length to be an LCD code is that C=(f(x)) where f is a self-reciprocal polynomial in Z4[X].