Long binary narrow-sense BCH codes are normal

Let C be the binary narrow-sense BCH code of length n=(2^m -1)/N and designed distance 2t+1, where m is the order of 2 module n. Using characters of finite fields and a theorem of Weil, and results of Vladut-Skorobogatov (1989) and Lang-Weil (1954) we prove that the code C is normal in the non-primitive case N>1 if 2^m⩾4(2tN)^4t+2, and in the primitive case N=1 if m⩾m₀, where the constant m₀ depends only on t