We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the “short code” of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results.

In particular, we prove quasi-NP-hardness of the following problems on $n$-vertex hyper-graphs:

* Coloring a 2-colorable 8-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors.

* Coloring a 4-colorable 4-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors.

* Coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{\Omega(1/\log\log\log n)}$ colors.

In each of these cases, the hardness results obtained are (at least) exponentially stronger than what was previously known for the respective cases. In fact, prior to this result, $(\log n)^{O(1)}$ colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for $O(1)$-colorable hypergraphs.

The fundamental bottleneck in obtaining coloring inapproximability results using the low- degree long code was a multipartite structural restriction in the PCP construction of Dinur- Guruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a “query doubling” method exploiting additional properties of the 8-query test. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form. The latter step involves extending the key algebraic ingredient of Dinur-Guruswami concerning testing binary Reed-Muller codes to the ternary alphabet.