An error model with asymmetric errors of limited magnitude is a good model for some multilevel flash memories. This paper is about constructions of codes correcting such errors. The main results are about codes correcting a single such error and codes of lengthmcorrecting all errors inm-1 or less positions.

Codes for error detection on aq-ary symmetric channel are studied. Whether a code is good or not for error detection (in the technical sense) depends on the structure of the code. For some combinations of the main parameters length, size, and minimum distance, all code are good and for some other combinations all are ugly (stronger than not good). The purpose of this paper is to give bounds on the parameters for codes that are not good for error detection. In particular, it is shown that if the minimum distance is below some bound, which depends on the length and size of the code as well as a lower bound on the number of codewords of minimum distance, then the code is ugly and, hence, not good for error detection.

Codes that can correct up totsymmetric errors and detect all unidirectional errors are studied. BOumlinck and van Tilborg gave a bound on the length of binary such codes. A generalization of this bound to arbitrary alphabet size is given. This generalized BOumlinck-van Tilborg bound, combined with constructions, is used to determine some optimal binary and ternary codes for correctingtsymmetric errors and detecting all unidirectional errors.

Bose and Lin introduced a class of systematic codes for detection of binary asymmetric errors. In this note, we describe a generalization to q-ary asymmetric error detecting codes. For these codes, the possible undetectable errors are characterized and the undetectable errors of minimum weight are determined