Peak-Shift Control Codes for the $L_1$ Metric (European Journal of Pure and Applied Mathematics, 2025)
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, \( \mathbb{N} \). 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.
Keywords: Recording codes; peak-shift correction; L_1 distance; asymmetric distance; elementary symmetric functions; constant weight codes