Constructions of low-degree and error-correcting epsilon-biased sets [article]

Amir Shpilka
2005 Electronic colloquium on computational complexity  
In this work we give two new constructions of -biased generators. Our first construction answers an open question of Dodis and Smith [DS05], and our second construction significantly extends a result of Mossel et al. [MST03]. In particular we obtain the following results: 1. We construct a family of asymptotically good binary codes such that the codes in our family are also -biased sets for an exponentially small . Our encoding and decoding algorithms run in polynomial time in the block length
more » ... f the code. This answers an open question of Dodis and Smith [DS05]. 2. We construct a degree k -biased generator, G : {0, 1} m → {0, 1} n , for every k = o(log n). For k constant we get that n = Ω(m/log(1/ )) k , which is nearly optimal. Our result also separates degree k generators from generators in NC 0 k , showing that the stretch of the former can be much larger than the stretch of the latter. This problem of constructing degree k generators was introduced by Mossel et al. [MST03] who gave a construction only for the case of degree 2 generators.
dblp:journals/eccc/ECCC-TR05-155 fatcat:6tgmhgd6crhspkjleqojgudfaa