(untitled) (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, \( N \). 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.