q-ary (1,k)-overlap-free codes with given restrictions

Two words u and v have a t-overlap if the length t prefix of u is equal to the length t suffix of v, or vice versa. A code $C$ is t-overlap-free if no two words u and v in $C$ (including $u=v$) have a t-overlap. A code of length n is said to be $(t_1,t_2)$-overlap-free if it is t-overlap-free for all t such that $1⩽t_1⩽t⩽t_2⩽n-1$. A $(1,n-1)$-overlap-free code of length n is called non-overlapping, which has applications in DNA-based data storage systems and frame synchronization. In this paper, we initialize the study for codes of length n which are simultaneously $(1,k)$-overlap-free and $(n-k,n-1)$-overlap-free, and establish lower and upper bounds for the size of balanced and error-correcting $(1,k)$-overlap-free codes.