  Motivated by the sequence reconstruction problem initiated by Levenshtein,
reconstruction codes were introduced by Cai \emph{et al}. to combat errors when
a fixed number of noisy channels are available. The central problem on this
topic is to design codes with sizes as large as possible, such that every
codeword can be uniquely reconstructed from any $N$ distinct noisy reads, where
$N$ is fixed. In this paper, we study binary reconstruction codes with the
constraint that every codeword is balanced, which is a common requirement in
the technique of DNA-based storage. For all possible channels with a single
edit error and their variants, we design asymptotically optimal balanced
reconstruction codes for all $N$, and show that the number of their redundant
symbols decreases from $\frac{3}{2}\log_2 n+O(1)$ to
$\frac{1}{2}\log_2n+\log_2\log_2n+O(1)$, and finally to
$\frac{1}{2}\log_2n+O(1)$ but with different speeds, where $n$ is the length of
the code. Compared with the unbalanced case, our results imply that the
balanced property does not reduce the rate of the reconstruction code in the
corresponding codebook.
