The gold standard for testing randomness is the Diehard suite of tests, which is some stuff that George Marsaglia put together in the 90s at Florida State. The maintenance of this suite leaves a little to be desired, and it seems that development ended abruptly, but the suite is still good. It is designed to work on 32 bit ints in binary files, though, so you'll have to do some work to make it fit, but the routines do all sorts of tests for randomness you might not think of (runs, high/low bit randomness, some serial correlation checks, and so forth). There are 16 tests in all.
At the very least, I suggest you download the Diehard source and read the file tests.txt to get an idea of the kinds of things you could do.
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