New constructions for $3$-free and $3^+$-free binary morphisms

We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary binary morphisms. Such morphisms map binary letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue-Morse word $\mathbf{t}$ gives a complementary morphism that is $3^+$-free and then $\alpha$-free for any real $\alpha>3$. We also describe the lengths of all prefixes of $\mathbf{t}$ that give cubefree complementary morphisms by a 4-state binary finite automaton. Next we show that cubefree complementary morphisms of length $k$ exist for all $k\ne\{3,6\}$. Moreover, if $k$ is not representable as $3\cdot2^n$, then the images of letters can be chosen to be factors of $\mathbf{t}$. In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.