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We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphswith $2^{(\log N)^{1/10-o(1)}}$ colors, where $N$ is the number of vertices.There has been much focus on hardness of hypergraph coloring recently.Guruswami, H{\aa}stad, Harsha, Srinivasan and Varma showed that it isquasi-NP-hard to color 2-colorable 8-uniform hypergraphs with$2^{2^{\Omega(\sqrt{\log\log N})}}$ colors. Their result is obtained bycomposing standard Label Cover with an inner-verifier based on Low Degree LongCode, using Reed-Muller code testing results by Dinur and Guruswami. Using adifferent approach, Khot and Saket constructed a new variant of Label Cover,and composed it with Quadratic Code to show quasi-NP-hardness of coloring2-colorable 12-uniform hypergraphs with $2^{(\log N)^c}$ colors, for some $c$around 1/20. Their construction of Label Cover is based on a new notion ofsuperposition complexity for CSP instances. The composition with inner-verifierwas subsequently improved by Varma, giving the same hardness result for8-uniform hypergraphs.  Our construction uses both Quadratic Code and Low Degree Long Code, andbuilds upon the work by Khot and Saket. We present a different approach toconstruct CSP instances with superposition hardness by observing that when thenumber of assignments is odd, satisfying a constraint in superposition is thesame as "odd-covering" the constraint. We employ Low Degree Long Code in orderto keep the construction efficient. In the analysis, we also adapt andgeneralize one of the key theorems by Dinur and Guruswami in the context ofanalyzing probabilistically checkable proof systems.

$2^{(\log N)^{1/10-o(1)}}$ Hardness for Hypergraph Coloring