Insertion/Deletion Detecting Codes and the Boundary Problem

Abstract

Insertion/deletion detecting codes were introduced by Konstantinidis et al. In this paper we define insertion/deletion detecting codes in a slightly different manner, and based on this definition, we introduce multiple deletion and multiple insertion detecting codes. It is shown that these codes, which are systematic, are optimal in the sense that there exists no other systematic multiple deletion (insertion) detecting codes with a better rate. One of the limitations of number-theoretic code constructions intended to correct insertion/deletion errors, e.g., the Levenshtein code, is that they require received codeword boundaries to be known in order to successfully decode. In literature, a number of schemes have been proposed to deal with this problem. We show how insertion/deletion detecting codes as presented in this paper can be used to improve and/or extend some of these schemes.

Randomized generation of error control codes with automata and transducers

“…The inverse of del 1 is ins 1 and is shown in Figure 1, where recall it results by simply exchanging the order of the two words in all the labels in del 1 . By statement 2 of the above remark, the del 1 -detecting codes are the same as the ins 1 -detecting ones, and the same as the (del 1 ∨ ins 1 )-detecting ones-this is shown in [22] as well. The method of using transducers to model channels is quite general and one can give many more examples of past channels as transducers, as well as channels not studied before.…”

“…The latter case is when sub 2 takes the transition (s, 0/1, t 1 ) or (s, 1/0, t 1 ), corresponding to one error, and then possibly (t 1 , 0/1, t 2 ) or (t 1 , 1/0, t 2 ), corresponding to a second error. id 2 : On input x, id 2 outputs a word that results by inserting and/or deleting at most 2 symbols in x. del 1 , ins 1 : considered in [22]. On input x, del 1 (resp., ins 1 ) outputs either x, or any word that results by deleting (resp., inserting) exactly one symbol in x, and then inserting a symbol at the end of x (resp., deleting the last symbol of x).…”

“…In Table 4, the entries for [N = 100, = 8, end = 01; er = del 1 ] correspond to the systematic code of [22], that is, the code consisting of all codewords ending with 01-any of the 64 possible 6-bit words can be used in the positions 1-6 of these codewords. In the same table, the entries for [N = 500, = 12, end = ε; er = ov] correspond to block solid codes of length 12.…”

Implementation of Code Properties via Transducers

“…A code property, or independence, [15], is a set P of languages for which there is n ∈ N ∪ {ℵ 0 } such that L ∈ P, if and only if L ∈ P, for all L ⊆ L with 0 < |L | < n. If L is in P then we say that L satisfies P. Thus, L satisfies P exactly when all nonempty subsets of L with less than n elements satisfy P. A language L ∈ P is called P-maximal, or a maximal P code, if L ∪ {w} / ∈ P for any word w / ∈ L. We note that every L satisfying P is included in a maximal P code [15]. As far as we know, all code related properties in the literature [4,6,8,11,15,23,28] are code properties as defined here. The focus of this work is on 3-independences that can also be viewed as independences with respect to a binary relation in the sense of [29].…”

“…We note that the transducer approach to defining error-detecting code properties is very powerful, as it allows one to model insertion and deletion errors, in addition to substitution errors-see Fig 2. Codes for such errors are actively under investigation-see [23], for instance.…”