Two Deletion Correcting Codes From Indicator Vectors

Construction of capacity achieving deletion correcting codes has been a baffling challenge for decades. A recent breakthrough by Brakensiek <italic>et al</italic>., 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 <inline-formula> <tex-math notation="LaTeX">$7\log ({n})+ {o}(\log ({n}))$ </tex-math></inline-formula> redundant bits to encode an n-bit message, which is closer to optimal than previous constructions. Moreover, the encoding and decoding algorithms have <inline-formula> <tex-math notation="LaTeX">${O}({n})$ </tex-math></inline-formula> time complexity.