If you have the Statistics and Machine Learning Toolbox, then you can generate n Rayleigh-distributed random numbers using the raylrnd function:
r = raylrnd(b,n,1);
Choose the parameter b to be equal to the mean you want times sqrt(2/pi).
The Rayleigh distribution has a minimum of zero, so if you want a non-zero minimum, you have criteria that cannot be met simultaneously: Rayleigh-distributed, and non-zero minimum. You'll have to give up one of those.
One possibility would be to add a constant K to your randomly chosen values (and subtract K from the mean value you generate). Then your distribution would have the shape of a Rayleigh distribution, offset by that constant.
