Two Deletion Correcting Codes from Indicator Vectors

TL;DR

This paper introduces a new binary two-deletion correcting code construction using indicator vectors, achieving near-optimal redundancy and extending Varshamov-Tenengolts techniques, with potential applications in DNA storage.

Contribution

The paper presents a novel code construction method based on indicator vectors, reducing redundancy and generalizing Varshamov-Tenengolts codes for two-deletion correction.

Findings

Achieves near-optimal redundancy of 7 log(n) bits

Generalizes Varshamov-Tenengolts code construction

Provides a practical approach for deletion correction

Abstract

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 near-optimal.