On the List Decodability of Insertions and Deletions

Synchronization Strings and Codes for Insertions and Deletions -- a Survey

“…We generally define the editdistance between two strings as the smallest number of insertions and deletions needed to convert one to another. The bounds presented in [40] show that binary codes by Bukh, Guruswami, and Håstad [10] can be list-decoded from a fraction ≈ 0.707 of insertions. Via a concatenation scheme used in [31] and [30], Hayashi and Yasunaga furthermore made these codes efficient.…”

“…Haeupler et al [38,37] gave upper and lower bounds on the maximum achievable rate of list-decodable insertion-deletion codes (or insdel codes for short) over any alphabet size q. Recent works of Wachter-Zeh [62] and Hayashi and Yasunaga [40] have studied list-decoding by providing Johnson-type bounds for synchronization codes that relate the minimum edit-distance of the code to its list decoding properties. We generally define the editdistance between two strings as the smallest number of insertions and deletions needed to convert one to another.…”

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

“…Let p 1 = d 2 . Then for each j ∈ {1, 2}, either u i ≥ 0 for all i ∈ [p j−1 + 1, p j ] or u i ≤ 0 for all i ∈ [p j−1 + 1, p j ], where p 0 = 1 and p 2 = n. Moreover,|u i | ≤ 1 for all i ∈ [n].Proof of Claim 5:For i ∈ [1, d 2 ], by(18), we haveu i = 0 or u i = x i − x ′ d2 . If x ′ d2 = 0, then u i ≥ 0 for all i ∈ [1, d 2 ]; if x ′ d2 = 1, then u i ≤ 0 for all i ∈ [1, d 2 ].…”

“…Thus, p 1 = d 2 satisfies the desired property. Note that |x λ2 − x ′ λ2 | ≤ 1 and |x i − x ′ d2 | ≤ 1.It is easy to see from(18) that |u i | ≤ 1 for all i ∈ [n], which proves Claim 5.Similar to Case (i), by (2) and by Claim 5, for j = 1, 2,|f j (x) − f j (x ′ )| ≤ n i=1 |u i |i j−1 ≤ n i=1 i j−1 < n j .…”

“…If x ′ d2 = 0, then u i ≥ 0 for all i ∈ [1, d 2 ]; if x ′ d2 = 1, then u i ≤ 0 for all i ∈ [1, d 2 ]. For i ∈ [d 2 + 1, n], by(18), we haveu i = 0 or u i = x λ2 − x ′ λ2 . If x ′ λ2 = 0, then u i ≥ 0 for all i ∈ [d 2 + 1, n]; if x ′ λ2 = 1, then u i ≤ 0 for all i ∈ [d 2 + 1, n].…”