In this paper we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where a data set is represented by an unordered set of $M$ sequences, each of length $L$ . Errors within that model are a loss of whole sequences and point errors inside the sequences, such as insertions, deletions and substitutions. We derive Gilbert-Varshamov lower bounds and sphere packing upper bounds on achievable cardinalities of error-correcting codes within this storage model. We further propose explicit code constructions than can correct errors in such a storage system that can be encoded and decoded efficiently. Comparing the sizes of these codes to the upper bounds, we show that many of the constructions are close to optimal.

中文翻译：

---

对 DNA 存储的集合进行编码

在本文中，我们研究了用于在合成脱氧核糖核酸 (DNA) 中存储数据的纠错代码。我们研究了一个存储模型，其中数据集由一组无序的 百万美元 序列，每个长度 $L$ . 该模型中的错误是整个序列的丢失和序列内部的点错误，例如插入、删除和替换。我们根据该存储模型中可实现的纠错码基数推导出 Gilbert-Varshamov 下界和球体填充上限。我们进一步提出了显式代码结构，以便在可以有效编码和解码的存储系统中纠正错误。将这些代码的大小与上限进行比较，我们表明许多结构接近最优。