Constructions and Bounds for q-Ary (1, k)-Overlap-Free Codes

A
$(1,k)$
-overlap-free code, motivated by applications in DNA-based data storage systems and synchronization between communication devices, is a set of words in which no prefix of length
$t$
of any word is the suffix of any word for every integer
$t$
such that
$1\leq t\leq k$
. A
$(1,n-1)$
-overlap-free code of length
$n$
is said to be non-overlapping. We provide a construction for
$q$
-ary
$(1,k)$
-overlap-free codes of length
$2k$
, which can be viewed as a generalization of the Zero Block Construction presented by Blackburn, Esfahani, Kreher and Stinson recently over a binary alphabet, and analyze the asymptotic behavior of their sizes. When
$n\geq 2k$
, an explicit general lower bound and an asymptotic lower bound for the size of an optimal
$q$
-ary
$(1,k)$
-overlap-free code of length
$n$
are presented. The exact value of the maximum size of
$q$
-ary (1, 2)-overlap-free codes of length
$n$
is determined for any
$n\geq 4$
, and a construction for
$q$
-ary
$(1,k)$
-overlap-free codes of length
$k+2$
is given.