On the generalized concatenated construction for the Nordstrom-Robinson code and the binary Golay code (Q2068822)

For an alphabet \(E_q =\{0, 1,\ldots, q-1\}\) of size \(q\), every \(C\subseteq E_q^n\) is a \(q\)-ary code and its denoted by \((n, N, d)_q,\) where \(N=|C|, \) \(d\) is the minimum (Hamming) distance. A linear code \(C\) with parameters \((n, N = q^k, d)_q\) is denoted by \([n, k, d]_q.\) Based on the known fact that Nordstrom-Robinson code and the binary Golay code are unique up to equivalence, this work shows that the nonlinear binary Nordstrom-Robinson code with parameters \(n = 16, N = 28, d = 6\) constructed by Nordstrom and Robinson is a generalized concatenated code of order 3. This is achieved using the code \(C\) based on the outer codes \(A = \{(0 0 0 0), (1 1 1 1)\}\), \(A_1 \cup V_1\) and \(A_1 \cup V_1\) and the inner code \(B = B_1 \cup B_2\); the code \(C_1\) based on the outer codes \(A_1\) and \(A_2\) and the inner code \(B_1\); the code \(C_2\) based on the outer codes \(V_1\) and \(V_2\) and the inner code \(B_2\). The same fact is shown for the binary extended perfect Golay code with parameters \(n = 24\), \(N = 212\), \(d = 8\) constructed in 1949 by Golay.