Can I request the code behind a research paper from the author?
Ask for the code. Please do, however, explain why you want it and what you intend to use it for. 

Personally, I would be quite happy if someone contacted me about my research, and would try to give all the necessary tools to recreate my data. Especially so if I am not currently working on a follow up piece.

That said, I've asked for parameters, codes, procedures etc. several times from the authors when the description in a paper has been vague. I've had a lot of different responses: Some have plain ignored me, others have given me everything I asked for, and yet others have only obliged when I've suggested that I might be willing to put them as co-authors if I built upon their code in a way that leads to a publication. Finally, some have refused my request. This has happened for a number of reasons, for example those discussed in this question.

If you ask for code and say that you just want to verify some of the conclusions of the authors or use it as a reference for your own implementation, you can most typically expect a refusal or no reply. This, at least, is the experience I've had.