We prove that for every constant c and = ( log n ) − c , there is no polynomial time algorithm that when given an instance of 3LIN with n variables where an (1 − ) -fraction of the clauses are satisfiable, finds an assignment that satisfies at least ( 2 1 + ) -fraction of clauses unless NP BPP . The previous best hardness using a polynomial time reduction achieves = ( log log n ) − c , which is obtained by the Label Cover hardness of Moshkovitz and Raz [J. ACM, 57(5), 2010] followed by the reduction from Label Cover to 3LIN of Hastad [J. ACM, 48(4):798--859, 2001].

Our main idea is to prove a hardness result for Label Cover similar to Moshkovitz and Raz where each projection has a linear structure. This linear structure of Label Cover allows us to use Hadamard codes instead of long codes, making the reduction more efficient. For the hardness of Linear Label Cover, we follow the work of Dinur and Harsha [SIAM J. Comput., 42(6):2452--2486, 2013] that simplified the construction of Moshkovitz and Raz, and observe that running their reduction from a hardness of the problem LIN (of unbounded arity) instead of the more standard problem of solving quadratic equations ensures the linearity of the resultant Label Cover.