For every positive integer n greater than 4 there is a set of Latin squares of order n such that every permutation of the numbers 1,…,n appears exactly once as a row, a column, a reverse row or a reverse column of one of the given Latin squares. If n is greater than 4 and not of the form p or 2p for some prime number p congruent to 3 modulo 4, then there always exists a Latin square of order n in which the rows, columns, reverse rows and reverse columns are all distinct permutations of 1,…,n, and which constitute a permutation group of order 4n. If n is prime congruent to 1 modulo 4, then a set of (n−1)∕4 mutually orthogonal Latin squares of order n can also be constructed by a classical method of linear algebra in such a way, that the rows, columns, reverse rows and reverse columns are all distinct and constitute a permutation group of order n(n−1).

An alternative proof is given of the existence of greatest lower bounds in the imbalance order of binary maximal instantaneous codes of a given size. These codes are viewed as maximal antichains of a given size in the infinite binary tree of 0–1 words. The proof proposed makes use of a single balancing operation within the same imbalance poset of codes of the same fixed size, instead of moving back and forth between posets of codes corresponding to two different code sizes using expansion and contraction, as in the previous proofs of the existence of glb. It also makes use of a new combinatorial characterization of the imbalance order.