Deletions and Insertions of the Symbol “0” and Asymmetric/Unidirectional Error Control Codes for the L Metric

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).