Peak-Shift Control Codes for the  $L_1$ Metric

This paper gives some theory and efficient design of $q$-ary block codes capable of controlling the single peak-shifts of one direction (left or right shift) of size $l$. This problem of controlling $l_{L}(l_{R})$-peak-shift is shown to be equivalent to the efficient design of some $L_{1}$ metric asymmetric error control codes over the natural alphabet, $\inat$. From the relations with the $L_{1}$ distance error control codes and constant weight codes, new improved upper and lower bounds on the size of the optimal single $l_{L}(l_{R})$-peak-shifts error correcting codes are given. Furthermore, some non-systematic code designs are also given. Decoding can be efficiently performed by algebraic means with the Extended Euclidean Algorithm.