Coding for the permutation channel with insertions, deletions, substitutions, and erasures

Codes in the Space of Multisets—Coding for Permutation Channels With Impairments

“…The idea is to set l large enough so that the first summand on the right-hand side of ( 30) is minimized, but small enough so that the second summand is still negligible compared to the first one. Observe the relation (29) and the second summand on the right-hand side of (30). From (29) we see that, as n → ∞ and q ∼ qn, the sum…”

“…The idea is to set l large enough so that the first summand on the right-hand side of (30) is minimized, but small enough so that the second summand is still negligible compared to the first one. Observe the relation (29) and the second summand on the right-hand side of (30). From (29) we see that, as n → ∞ and q ∼qn, the sum |△ q−1 n | = (see (34)), and since it has linearly many summands, there must exist λ ∈ (0, 1) such that q λn n−1 λn grows exponentially in n with the same exponent.…”

“…. This gives the first summand in the upper bound (30). The second summand is simply the cardinality of the set of all vectors in △ q−1 n having less than l non-zero coordinates, which is certainly an upper bound on M ′′ .…”

“…This work was supported by the Singapore Ministry of Education (MoE) Tier 2 grant "Network Communication with Synchronization Errors: Fundamental Limits and Codes" (Grant number R-263-000-B61-112). Part of the work was presented at the 2017 IEEE International Symposium on Information Theory (ISIT) [30].…”

Marker Codes Using the Decoding Based on Weighted Levenshtein Distance in the Presence of Insertions/Deletions

“…Loss of synchronization due to the imperfect of the sampling clocks may cause catastrophic consequences with the variable length and enormous substitutions, which are of great interest in the communication systems [1][2][3][4][5]. Channels with errors caused by the loss synchronization have memory, and the techniques designed for memoryless channels can seldom be employed directly [6][7][8][9][10][11][12].…”