Optimal Systematic t-Deletion Correcting Codes

List-decodable Codes for Single-deletion Single-substitution with List-size Two

“…Moreover, we have x λ1 = x ′ λ1 and x λ2 = x ′ λ2−1 because of the substitution error. According to (11), this case can be illustrated by Fig. 3.…”

“…By Remark 3, we have wt(x) = wt(x ′ ) and x d1 = x ′ d2 . Then by (11) or Fig. 3, and through a cancelling process similar to Case (i), we can obtain 0…”

“…The best known t-edit correcting codes for t ≥ 2 were given by Sima et al, which have redundancy 4t log n + o(log n) [11]. The method in [11] was improved by the authors in [15], which gave a construction of t-deletion s-substitution correcting codes with redundancy (4t+ 3s) log n+ o(log n). A family of single-deletion single-substitution correcting binary codes with redundancy 6 log n + 8 was constructed in [13].…”

t-Deletion-1-Insertion-Burst Correcting Codes

“…The general problem of correcting t arbitrary deletions was considered in a series of works [1], [7], [8], [16]- [18]. The state-of-the-art result of them is the one from [17], which constructed a t-deletion-correcting code with redundancy 4t log n + o(log n).…”

A Quaternary Code Correcting a Burst of at Most Two Deletion or Insertion Errors in DNA Storage

“…However, designing q -ary VT codes that can correct multiple deletion or insertion errors has been an interesting problem [ 17 ]. Recently, there were some works [ 18 , 19 , 20 , 21 ] focused on code design to correct exact multiple errors but the efficient design for q -ary codes (or even quaternary codes) that can correct a burst of at most b deletion or insertion errors is still an open problem. The authors of [ 22 ] proposed a non-binary code correcting at most two consecutive deletions with redundancy .…”