This paper 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 0-errors) is shown to be equivalent to the efficient design of$L_{1}$metric asymmetric error control codes over the natural alphabet,${\mathbf{I}}\!{\mathbf{I}}\!\!{\mathbf{N}}$. In this way, it is shown that the$t 0$-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$L_{1}$distance error control codes, new improved bounds are given for the optimal$t 0$-error correcting codes. Optimal non-systematic code designs are given. Decoding can be efficiently performed by algebraic means using the Extended Euclidean Algorithm (EEA).

In this paper the theory and design of codes capable of correcting t insertion/deletion of the symbol 0 in each and every bucket of zeros (i. e., zeros in between two consecutive ones) are studied. It is shown that this problem is related to the zero error capacity achieving codes in limited magnitude error channel. Close to optimal non-systematic code designs and the encoding/decoding algorithms are described.

This paper 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 con-trolling 0-deletions and/or 0-insertions (referred to as 0-errors) is shown to be equivalent to the efficient design of L1metric asymmetric error control codes over the natural alphabet,IN. Optimal systematic code designs are given. In particular, for all $t,k \in {\mathbb{I}}\mathbb{N}$, a recursive method is presented to encode k information bits into efficient systematic t Symmetric 0-Error Correcting, (t + 1) Symmetric 0-Error Detecting and All Unidirectional 0-Error Detecting (t-Sy0EC/(t+1)-Sy0ED/AU0ED) codes of length\begin{equation*}n \leq k + t\;{\text{lo}}{{\text{g}}_2}\;k + o(t\;{\text{log}}\;n)\end{equation*}as $n \in {\mathbb{I}}\mathbb{N}$ increases. Decoding can be efficiently performed by algebraic means using the Extended Euclidean Algorithm (EEA).

This paper gives some theory and efficient design of binary block codes capable of correcting 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 correcting 0-deletions and/or 0-insertions (referred to as 0-errors) is shown to be equivalent to the efficient design of some L1metric asymmetric error control codes over the natural alphabet, ℕ. In particular, it is shown that t 0-insertion correcting codes are actually capable of correcting t 0-errors, detecting (t+1) 0-errors and, simultaneously, detecting all occurrences of only 0-deletions or only 0-insertions in every received word (briefly, they are t-Sy0EC/(t + 1)-Sy0ED/AU0ED codes). From the relations with the L1distance error control codes, new improved bounds are given for the optimal t 0-error correcting codes. In addition, some optimal non-systematic code designs are also given. Decoding can be efficiently performed by algebraic means with the Extended Euclidean Algorithm.