TAOCP 1.3.3 Exercise 33
Working
Section 1.3.3: Applications to Permutations
Exercise 33. [M33] If $m = 2^{2^l}$ and $n = 2^{2l+1}$, show how to construct sequences of permutations $(\alpha_{j1}, \alpha_{j2}, \ldots, \alpha_{jn}; \beta_{j1}, \beta_{j2}, \ldots, \beta_{jn})$ for $0 \le j < m$ with the following "orthogonality" property:
$$ \alpha_{i1}\beta_{j1}\alpha_{i2}\beta_{j2}\cdots\alpha_{in}\beta_{jn} = \begin{cases} (1,2,3,4,5), & \text{if } i = j;\ (), & \text{if } i \ne j. \end{cases} $$
Each $\alpha_{jk}$ and $\beta_{jk}$ should be a permutation of ${1, 2, 3, 4, 5}$.
Verified: no
Solve time: 6m48s
Working