Binary encoding with short bit sequences

Let row $i$ consist of any sequence of length $p_i$ that includes both $0$ and $1$ (for definiteness, say, $0^{p_i-1}1$), where $p_i$ denotes the $i$th prime. Then all columns eventually occur by the Chinese remainder theorem, and the longest sequence has period $p_n=O(n\log n)$ by Chebyshev's theorem.

We can improve the construction to get essentially optimal sequences as follows. If we want only sequences of length $\le m$, then for each prime $p\le m$, we can take up to $\lfloor\log_2p\rfloor$ rows of length $p$ such that all columns restricted to these rows occur, using e.g. the binary encoding as in the question. (Slightly better, we could take $\lfloor k\log_2p\rfloor$ sequences of length $p^k$, where $p^k$ is the maximal power of $p$ such that $p^k\le m$, but this would not change the asymptotics.) So the sequences can look like this:

01
010
01010
00110
0101010
0011001
01010101010
00110011001
00001111000
0101010101010
0011001100110
0000111100001
01010101010101010
00110011001100110
00001111000011110
00000000111111110
0101010101010101010
0011001100110011001
0000111100001111000
0000000011111111000
...

Then all columns still occur by the Chinese remainder theorem. How large $m$ we need to get $n$ rows? This construction gives us up to $\sum_{p\le m}\lfloor\log_2p\rfloor$ rows, hence we need $$n\le\sum_{\substack{p\le m\\\text{$p$ prime}}}\lfloor\log_2p\rfloor=\Omega(m)$$ using Chebyshev’s theorem again, hence it suffices to take $$m=O(n).$$ More precisely, the prime number theorem implies $$\sum_{\substack{p\le m\\\text{$p$ prime}}}\lfloor\log_2p\rfloor\sim\frac m{\ln2},$$ whence it suffices to take $$m=(\ln 2+o(1))n.$$ This bound is asymptotically optimal: if we can get all $2^n$ columns using $n$ sequences of lengths $m_1,\dots,m_n\le m$, then $$2^n\le\operatorname{lcm}\{m_1,\dots,m_n\}\le\operatorname{lcm}\{1,\dots,m\}=e^{(1+o(1))m}$$ using the prime number theorem again, hence $m$ is at least $(\ln2+o(1))n$.

There is nothing special about binary in the argument above: if we fix any finite alphabet $\Sigma$ of size $|\Sigma|\ge2$, then for any $n$, we can construct $n$ sequences over $\Sigma$ of length at most $$(\ln|\Sigma|+o(1))n$$ such that every column occurs when they are extended periodically, and this is asymptotically optimal.