Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with $n$ variables and $m$ clauses, there is a value of $m = Omega(n)$ beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when $m/n = omega(1)$). Intuitively, strong refutation should become easier as the clause density $m/n$ grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as $k$-SAT and $k$-XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, $m/n ge widetilde O(n^{k/2-1})$, and the clause density at which instances become unsatisfiable with high probability, $m/n = omega (1)$. In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random $k$-XOR instances with clause density $m/n ge widetilde O(n^{(k/2-1)(1-delta)})$ in time $exp(widetilde O(n^{delta}))$ or in $widetilde O(n^{delta})$ rounds of the sum-of-squares hierarchy, for any $delta in [0,1)$ and any integer $k ge 3$. Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at $delta = 0$, and brute-force refutation at the satisfiability threshold when $delta = 1$. We also leverage our $k$-XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors.
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