A binary vector of length \(n\) which is \(l\)-balanced with the greatest number of binary vectors (Q1358560)

E. P. Lipatov suggested in 1987 that the following distance \(M\) on the set \(\{0,1\}^n\) could replace the Hamming distance: \(M(\alpha,\beta)= \max_{1\leq k_1\leq k_2\leq n}|\sum^{k_2}_{p=k_1}(\alpha_p- \beta_p)|\). Two vectors \(\alpha,\beta\in\{0,1\}^n\) are said to be \(l\)-balanced provided that \(M(\alpha,\beta)\leq l\). The author proves that the vector \((1,0,1,0,\dots)\) of length \(n\) is \(l\)-balanced with the largest number of vectors in \(\{0,1\}^n\).