Efficient Systematic Deletions/Insertions of 0’s Error Control Codes

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 L
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metric 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).