On q-ary codes correcting all unidirectional errors of a limited magnitude

We consider codes over the alphabet Q = {0, 1,. .. , q − 1} intended for the control of unidirectional errors of level ℓ. That is, the transmission channel is such that the received word cannot contain both a component larger than the transmitted one and a component smaller than the transmitted one. Moreover, the absolute value of the difference between a transmitted component and its received version is at most ℓ. We introduce and study q-ary codes capable of correcting all unidirectional errors of level ℓ. Lower and upper bounds for the maximal size of those codes are presented. We also study codes for this aim that are defined by a single equation on the codeword coordinates (similar to the Varshamov-Tenengolts codes for correcting binary asymmetric errors). We finally consider the problem of detecting all unidirectional errors of level ℓ.