The problem of reliable communication over the multiple-access channel (MAC) with states is investigated. We propose a new coding scheme for this problem which uses quasi-group codes (QGC). We derive a new computable single-letter characterization of the achievable rate region. As an example, we investigate the problem of doubly-dirty MAC with modulo-$4$ addition. It is shown that the sum-rate $R_1+R_2=1$ bits per channel use is achievable using the new scheme. Whereas, the natural extension of the Gel'fand-Pinsker scheme, sum-rates greater than $0.32$ are not achievable.