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Motivated by the sequence reconstruction problem initiated by Levenshtein, reconstruction codes were introduced by Cai 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 fundamental 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 3 2 log 2 n + O (1) to 1 2 log 2 n + log 2 log 2 n + O (1) , and finally to 1 2 log 2 n + 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we first generalize the polycyclic codes over finite fields to polycyclic codes over the mixed alphabet Z 2 Z 4 , and we show that these codes can be identified as Z 4 [ x ] -submodules of R α , β with R α , β = Z 2 [ x ] / ⟨ t 1 (x) ⟩ × Z 4 [ x ] / ⟨ t 2 (x) ⟩ , where t 1 (x) and t 2 (x) are monic polynomials over Z 2 and Z 4 , respectively. Then we provide the generator polynomials and minimal generating sets for this family of codes based on the strong Gröbner basis. In particular, under the proper defined inner product, we study the dual of Z 2 Z 4 -additive polycyclic codes. Finally, we focus on the characterization of the Z 2 Z 4 -MDSS and MDSR codes, and as examples, we also present some (almost) optimal binary codes derived from the Z 2 Z 4 -additive polycyclic codes. [ABSTRACT FROM AUTHOR]

Mixed alphabet codes are the generalizations of the classical linear codes over finite fields and rings. In this letter we introduce a valid method of constructing FpR-additive codes, where p is a prime number, and R = Fp + uFp with u2 = 0. These codes can be viewed as a combination of irreducible cyclic codes over Fp and trace codes over R. In particular, by using the Gray map, part of the obtained image codes meets the Greismer bound with equality, and can be applied to construct new secret sharing scheme. [ABSTRACT FROM AUTHOR]

In this paper, two different Gray-like maps from $Z_{p}^\alpha \times Z_{p^{k}}^\beta $ , where $p$ is prime, to $Z_{p}^{n}$ , $n={\alpha +\beta p^{k-1}}$ , denoted by $\varphi $ and $\Phi $ , respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_{p} Z_{p^{k}}$ -additive code, and $C^\bot $ is its dual, then the weight enumerators of the image $p$ -ary codes $\varphi (C)$ and $\Phi (C^\bot)$ are formally dual. This is a partial generalization of [D. S. Krotov, On $Z_{2^{k}}$ -dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532–1537], and the result is generalized to odd characteristic $p$ and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed $Z_{p} Z_{p^{2}}\ldots Z_{p^{k}}$ alphabet is given. [ABSTRACT FROM AUTHOR]

In this work, we study a family of self-dual four circulant codes and a family of self-dual four negacirculant codes, and the exact counting formula is derived for these families of codes. In addition, we prove that asymptotically good self-dual four circulant codes and negacirculant codes over finite fields exist, and both of them satisfy a modified Gilbert-Varshamov bound on the relative minimum distance with asymptotic rate 1 2. [ABSTRACT FROM AUTHOR]