رموز التحكم لعمليات حذف وإدراج الرمز صفر وتصحيح خطأ غير متماثل / أحادي الاتجاه لمقياس L1

Abstract

This work gives some theory and efficient design of binary block codes capable of controlling the deletions of the symbol “0” (referred to as 0-deletions) and/or the insertions of the symbol “0” (referred to as 0-insertions). This problem of controlling 0-deletions and/or 0-insertions (referred to as symmetric 0-errors) is shown to be equivalent to the efficient design of some L1 metric asymmetric error control codes over the natural alphabet, N. In particular, it is shown that t 0-insertion correcting codes are actually capable of controlling much more; namely, they can correct t 0-errors, detect (t + 1) 0-errors and, simultaneously, detect all occurrences of only 0-deletions or only 0-insertions in every received word (briefly, they are t-Symmetric 0-Error Correcting/(t +1)-Symmetric 0-Error Detecting/All Unidirectional 0- Error Detecting (t-Sy0EC/(t + 1)-Sy0ED/AU0ED codes). From the relations with the L1 distance error control codes, new improved bounds are given for the optimal t 0-error correcting codes. Optimal non-systematic and systematic code designs are given. In particular, for all t, k ∈ N, a recursive method is presented to encode k information bits into efficient systematic t-Sy0EC/(t +1)-Sy0ED/AU0ED codes of length n ≤ k +t log2 k +o(t logn) as n ∈ N increases. Decoding can be efficiently performed by algebraic means using the Extended Euclidean Algorithm (EEA).