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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.

Quasi-polycyclic (QP for short) codes over a finite chain ring $R$ are a generalization of quasi-cyclic codes, and these codes can be viewed as an $R[x]$-submodule of $\mathcal{R}_m^{\ell}$, where $\mathcal{R}_m:= R[x]/\langle f\rangle$, and $f$ is a monic polynomial of degree $m$ over $R$. If $f$ factors uniquely into monic and coprime basic irreducibles, then their algebraic structure allow us to characterize the generator polynomials and the minimal generating sets of 1-generator QP codes as $R$-modules. In addition, we also determine the parity check polynomials for these codes by using the strong Gr\"{o}bner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.

We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-$1$ balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$-fold packing with minimum distance $2$. Secondly, for every admissible length starting from $n=20$, we show the existence of $4$-ary codes with parameters of shortened $1$-perfect codes that cannot be obtained by shortening a $1$-perfect code. Keywords: Hamming graph, multifold packings, multiple coverings, perfect codes.