Long binary narrow-sense BCH codes are normal
The covering radius of a binary code \(C\) of length \(n\) is the smallest integer \(R\) such that the union of the spheres of radius \(R\) around the codewords covers the whole space \(F_2^n\). Let \(C_a^{(i)}\) denote the set of codewords of \(C\) whose \(i\)th component equals \(a\), \(a=0,1\). The code \(C\) is normal if for some \(i\), \(d(x,D_0^{(i)})+d(x,C_1^{(i)})\leq2R+1\) for all \(x\in F_2^n\) where \(d(x,y)\) denotes the Hamming distance between \(x,y\in F_2^n\). Let \(C\) be a binary narrow-sense BCH code of designed distance \(2t+1\) and length \(n=(2^m-1)/h\) where \(m\) is the order of 2 modulo \(m\). The authors show that this code is normal in the case \(h>1\) if \(2^m\geq4(2th)^{4t+2}\) and in the case \(h=1\) if \(m\geq m_0\), where \(m_0\) only depends on \(t\). The proofs are based upon results on character sums over finite fields.
scientific article; zbMATH DE number 978148