Construction of capacity achieving deletion correcting codes has been a baffling challenge for decades. A recent breakthrough by Brakensiek et al., alongside novel applications in DNA storage, have reignited the interest in this longstanding open problem. In spite of recent advances, the amount of redundancy in existing codes is still orders of magnitude away from being optimal. In this paper, a novel approach for constructing binary two-deletion correcting codes is proposed. By this approach, parity symbols are computed from indicator vectors (i.e., vectors that indicate the positions of certain patterns) of the encoded message, rather than from the message itself. Most interestingly, the parity symbols and the proof of correctness are a direct generalization of their counterparts in the Varshamov-Tenengolts construction. Our techniques require $7\log ({n})+ {o}(\log ({n}))$ redundant bits to encode an n-bit message, which is closer to optimal than previous constructions. Moreover, the encoding and decoding algorithms have ${O}({n})$ time complexity.

中文翻译：

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指示向量中的两个删除校正码

几十年来，实现删除纠正代码的能力建设一直是一个令人困惑的挑战。Brakensiek 最近的突破等. 以及 DNA 存储方面的新应用，重新点燃了人们对这个长期存在的开放问题的兴趣。尽管最近取得了进展，但现有代码中的冗余量仍与最佳状态相差几个数量级。在本文中，提出了一种构造二进制二删改正码的新方法。通过这种方法，奇偶校验符号是根据编码消息的指示向量（即，指示某些模式的位置的向量）计算的，而不是根据消息本身计算的。最有趣的是，奇偶校验符号和正确性证明是它们在 Varshamov-Tenengolts 构造中对应物的直接概括。我们的技术需要 $7\log ({n})+ {o}(\log ({n}))$ 冗余位来编码 n 位消息，这比以前的结构更接近最优。此外，编码和解码算法有 ${O}({n})$ 时间复杂度。