It was known that the empirical spectral distribution of random matrices constructed from binary linear codes of increasing length converges to the Marchenko-Pastur law as long as the dual distance of the codes is at least 5, and the condition of the dual distance 5 is optimal because there are binary linear codes of dual distance 4 that do not satisfy this property. In this article, we push this result a little further: we show that a Gaussian central limit theorem holds for the linear spectral statistics associated with such random matrices from binary linear codes of increasing length when the dual distance is at least 7. We also show that the condition of dual distance 7 is optimal as there are binary linear codes of dual distance 6 that do not satisfy this property. This result can be interpreted as that pseudorandom sequences constructed from long binary linear codes of dual distance 7 in general satisfy a more stringent pseudorandom test than those from binary linear codes of dual distance 5.