a transversal. In 1975, S. K. Stein and Brualdi conjectured that, when n is even, every n-by-n Latin square has a partial transversal of size n−1. A more general conjecture of Stein is that a transversal of size n−1 exists not only in Latin squares but also in any n-by-n array of n symbols, as long as each symbol appears exactly n times. Some weaker versions of these conjectures have been proved: Every n-by-n Latin square has a partial transversal of size 2n/3. Every n-by-n Latin square has a partial transversal of size n − sqrt(n). Every n-by-n Latin square has a partial transversal of size n − 11 log(n). Algorithms For small squares it is possible to generate permutations and test whether the Latin square property is met. For larger squares, Jacobson and Matthews' algorithm allows sampling from a uniform distribution over the space of n × n Latin squares. Applications Statistics and mathematics In the design of experiments, Latin squares are a special case of row-column designs for two blocking factors. In algebra, Latin squares are related to generalizations of groups; in particular, Latin squares are characterized as being the multiplication tables (Cayley tables) of quasigroups. A binary operation whose table of values forms a Latin square is said to obey the Latin square property. Error correcting codes Sets of Latin squares that are orthogonal to each other have found an application as error correcting codes in situations where communication is disturbed by more types of noise than simple white noise, such as when attempting to transmit broadband Internet over powerlines. Firstly, the message is sent by using several frequencies, or channels, a common method that makes the signal less vulnerable to noise at any one specific frequency. A letter in the message to be sent is encoded by sending a series of signals at different frequencies at successive time intervals. In the example below, the letters A to L are encoded by sending signals at four different frequencies, in four time slots. The letter C, for instance, is encoded by first sending at frequency 3, then 4, 1 and 2. The encoding of the twelve letters are formed from three Latin squares that are orthogonal to each other. Now imagine that there's added noise in channels 1 and 2 during the whole transmission. The letter A would then be picked up as: In other words, in the