  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,
$\mathbb{N}$. 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
($t$-Sy$0$EC/$(t+1)$-Sy$0$ED/AU$0$ED) codes). From the relations with the
$L_{1}$ distance, optimal systematic code designs are given. In general, for
all $t,k\in\mathbb{N}$, a recursive method is presented to encode $k$
information bits into efficient systematic $t$-Sy$0$EC/$(t+1)$-Sy$0$ED/AU$0$ED
codes of length $$ n\leq k+t\log_{2}k+o(t\log n) $$ as $n\in\mathbb{N}$
increases. Decoding can be efficiently performed by algebraic means using the
Extended Euclidean Algorithm (EEA).
