this paper, we propose improvements to a naive simulate and reject procedure for generating r \Theta c tables under quasi-independence for an arbitrary pattern of fixed cells. Although some of the algorithmic improvements are described for generating under QI for the off-diagonal cells of a square table, the ideas are applicable to other patterns of fixed cells. Apart from complete enumeration, which is only viable for small tables, the simulate and reject procedure is currently the only method for generating independent tables from the exact null distribution under QI. Our improvements to the naive procedure greatly increase its efficiency. Smith, McDonald and Forster (1994) discuss another method for generating tables under QI using a Gibbs sampling approach, based on theoretical results in Forster, McDonald and Smith (1994). However, the generated tables are not necessarily independent and are only realizations from an approximation to the exact null distribution. When using a single Markov chain, the observed table is the obvious starting value. For multiple chains, obtaining other starting values with the same sufficient statistics for the nuisance parameters as the observed data is problematic. A possible solution is to generate a small number of independent starting values using the simulate and reject algorithms proposed. Acknowledgements