Get a fixed-size family of k-element subsets
In general, what you are looking for is a combinatorial number system (of degree $t$) , i.e., a correspondece between $\mathbb{N}$ and all $t$-combinations.

There is a large number of algorithms for $t$-combination generation from the last (at least) 50 years. In particular, the operations you are interested in are typically called rank and unrank.

For much more than what the Wikipedia article contains at the time of writing, you can have a look at say the book of Knuth [1] or the one by Kreher and Stinson [2]. Knuth's book contains pseudocode that is straightforward to write in your favorite language, and the nook of Kreher and Stinson is accompanied by source code (in C) that can be found online as well.



[1] Knuth, Donald E. The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1. Pearson Education India, 2011.

[2] Kreher, Donald L., and Douglas R. Stinson. Combinatorial algorithms: generation, enumeration, and search. Vol. 7. CRC press, 1998.