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 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 7log(n)+o(log(n) redundant bits to encode an n-bit message, which is near-optimal

Practical Wireless Network Coding and Decoding Methods for Multiple Unicast Transmissions

We propose a simple yet effective wireless network coding and decoding technique. It utilizes spatial diversity through cooperation between nodes which carry out distributed encoding operations dictated by generator matrices of linear block codes. For this purpose, we make use of greedy codes over the binary field and show that desired diversity orders can be flexibly assigned to nodes in a multiple unicast network, contrary to the previous findings in the literature. Furthermore, we present the optimal detection rule for the given model that accounts for intermediate node errors and suggest a network decoder using the sum-product algorithm. The proposed sum-product detector exhibits near optimal performance.Comment: 6 pages, 5 figures, Submitted to WCNC 2012, IEEE Wireless Communication and Networking Conferenc

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 $et~al$., 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