Given $n$ binary sequences $s_i$ ($1\le i\le n$) with common period $T$. Let $s_i(t_i)$ denote the sequence obtained by cyclically shifting $s_i$ for $t_i$ bits where $t_i\in[0,T-1]$. Consider the following decision problem. Do there exist $t_i$'s such that for each $t\in[0,T-1]$, there exists only one single shifted sequence, whose bit is $1$ at $t$. Is this problem NP-hard?

The background of the problem is below. We want to design a code for each of the $n$ users in the way that for user $i$ with code $s_i$, it transmits its packet in slot $t$ if $s_i(t)=1$. We want to check whether a given set of codes can ensure that even the users are not time-synchronized with arbitrarily drifted local clocks, each of them can successfully transmit a packet, given that if two or more users transmit at the same slot, none of them succeeds.