Reconstruction Codes for DNA Sequences with Uniform Tandem-Duplication Errors

DNA as a data storage medium has several advantages, including far greater data density compared to electronic media. We propose that schemes for data storage in the DNA of living organisms may benefit from studying the reconstruction problem, which is applicable whenever multiple reads of noisy data are available. This strategy is uniquely suited to the medium, which inherently replicates stored data in multiple distinct ways, caused by mutations. We consider noise introduced solely by uniform tandem-duplication, and utilize the relation to constant-weight integer codes in the Manhattan metric. By bounding the intersection of the cross-polytope with hyperplanes, we prove the existence of reconstruction codes with greater capacity than known error-correcting codes, which we can determine analytically for any set of parameters.Comment: 11 pages, 2 figures, Latex; version accepted for publicatio

Sidon sets for linear forms

Let $\varphi(x_1,\ldots, x_h) = c_1 x_1 + \cdots + c_h x_h$ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $\varphi$-Sidon set if, for all $h$-tuples $(a_1,\ldots, a_h) \in A^h$ and $(a'_1,\ldots, a'_h) \in A^h$, the relation $\varphi(a_1,\ldots, a_h) = \varphi(a'_1,\ldots, a'_h)$ implies $(a_1,\ldots, a_h) = (a'_1,\ldots, a'_h)$. There exist infinite Sidon sets for the linear form $\varphi$ if and only if the set of coefficients of $\varphi$ has distinct subset sums. In a normed vector space with $\varphi$-Sidon sets, every infinite sequence of vectors is asymptotic to a $\varphi$-Sidon set of vectors. Results on $p$-adic perturbations of $\varphi$-Sidon sets of integers and bounds on the growth of $\varphi$-Sidon sets of integers are also obtained.Comment: Minor changes and improvements; 16 page