© 2020 IEEE.Flash memory devices are winning the competition for storage density against magnetic recording devices. This outcome results from advances in physics that allow storage of more than one bit per cell, coupled with advances in signal processing that reduce the effect of physical instabilities. Constrained codes are used in storage to avoid problematic patterns. Recently, we introduced binary symmetric lexicographically-ordered constrained codes (LOCO codes) for data storage and transmission. This paper introduces simple constrained codes that support non-binary physical gates in multi, triple, quad, and the currently-in-development penta-level cell (M/T/Q/P-LC) Flash memories. The new codes can be easily modified if problematic patterns change with time. These codes are designed to mitigate inter-cell interference, which is a critical source of error in Flash devices. The new codes are called q-ary asymmetric LOCO codes (QA-LOCO codes), and the construction subsumes codes previously designed for single-level cell (SLC) Flash devices (ALOCO codes). QA-LOCO codes work for a Flash device with any number, q, of levels per cell. For q ≥ 4, we show that QA-LOCO codes can achieve rates greater than 0.95log2 q information bits per coded symbol. Capacity-achieving rates, affordable encoding-decoding complexity, and ease of reconfigurability support the growing improvement of M/T/Q/P-LC Flash memory devices, as well as lifecycle management as the characteristics of these devices change with time.