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This paper gives some theory and efficient design of binary block systematic 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). The problem of controlling 0-deletions and/or 0-insertions (referred to as 0-errors) is known to be equivalent to the efficient design of L 1 metric asymmetric error control codes over the natural alphabet, N. So, t 0-insertion correcting codes can actually 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, optimal systematic code designs are given. In general, 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 lengthas n ∈ N increases. Decoding can be efficiently performed by algebraic means using the Extended Euclidean Algorithm (EEA).