Covariate imbalance and restricted randomization

Dear all,

I'm involved in a cluster-randomized clinical trial with 20 clusters and 2 treatment arms, intervention and control. We would like to perform restricted randomization on the 20 clusters eliminating all randomization schemes that do not have exactly 10 clusters each assigned to either intervention and control and which are not balanced in terms of important continuous co-variates (such as education and health service access). However, I am slightly concerned that if we specify too restrictive balancing criteria (e.g. no more than 2% difference in education levels between control and intervention), we may potentially eliminate quite a lot of randomization schemes.

For example, there are 18475 ways of choosing 10 out of 20 clusters for intervention. However, if we require education levels to be balanced at the 2% level, this reduces to just 30,000 or so schemes, which seems to be a relatively low number from a combinatorial point of view. Does anyone know of any published research on the appropriate level of restriction to specify for restricted randomization? I would intuitively expect, for example that if we get down to TOO few randomization schemes (100 or so) we will actually INTRODUCE bias in co-variates we are not restricting on incl. unobserved co-variates because "the level of randomness" in our choice of allocation scheme has decreased substantially.

Thanks in advance for any help you may offer on this topic!