Several physical effects that limit the reliability and performance of Multilevel Flash memories Induce errors that have low magnitude and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions for such channels when the number of errors is bounded by t. The construction uses known codes for symmetric errors over small alphabets to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. An extension of the construction is proposed to include systematic codes as a benefit to practical implementation.