The Reed-Muller code \(R(1,7)\) is normal (Q1364212)

Let \(C\) be an \([n,k]\) binary linear code with covering radius \(R\), and let \(C^{(i)}\) be the subcode consisting of all codewords with a zero in the \(i\)th position. The code \(C\) is said to be normal if there exists an \(i\) such that \[ d(x,C^{(i)}) +d(x,C \setminus C^{(i)}) \leq 2R+1 \quad\text{for all} \quad x\in GF(2)^n. \] The main result is that the first order Reed-Muller code of length 128, RM(1,7), is normal. Let \(t[n,k]\) be the smallest covering radius of any \([n,k]\) code. The normality of RM(1,7) leads to an improvement of the best upper bound on \(t[n,8]\). The new bound is \(t[n,8]\leq \lfloor {n-16\over 2} \rfloor\) for \(n\geq 127\).