Long Code testing is a fundamental problem in the area of propertytesting and hardness of approximation.Long Code is a function of the form f(x)=xi for some index i.In the Long Code testing, the problem is, given oracle access to acollection of Boolean functions, to decide whether all the functionsare the same Long Code, or cross-influences of any two functions aresmall.In this paper, we study the following problem:How small the soundness s of the Long Code test with perfectcompleteness can be by using non-adaptive q queries?We give a Long Code test with s=(2q+3)2q , where q is of the form2k−1 for any integer 2">k2.Our test is a ``noisy'' version of Samorodnitsky-Trevisan's Hyper Graphlinearity test with suitably chosen noise distribution.To bound the soundness, we use Invariance-Principle style analysisin the spirit of O'Donnell and Wu (STOC 2009).

Previously, H\r{a}stad and Khot (Theory of Computing, 2005) have showns=24q2q for infinitely many q.Chen (RANDOM 2009) has improved this to s=q32q for infinitely manyq with ``adaptive'' queries.As for the Long Code test with ``almost'' perfect completeness,Trevisan and Samorodnitsky (SICOMP, 2009) have showns=2q2q (or even (q+1)2q for infinitely many q).Austrin and Mossel (Computational Complexity) have improved this to s=(q+o(q))2q (or even (q+4)2q assuming the famous HadamardConjecture) for any q.