The Hamming(7,4) Code

Educação Superior::Ciências Exatas e da Terra::MatemáticaHamming(7,4) is a single-error correcting code that uses a 7-bit codeword to transmit four bits of data. The sender computes three parity bits for each 4-bit data word, assembles the data and parity bits into a 7-bit codeword, and transmits this codeword. The receiver computes three parity check bits from the received 7-bit word. If no error occurred in transmission, all three parity check bits will be zero. If a single bit has been changed in transmission, the value of the three parity bits (interpreted as a 3-bit binary number) will indicate the position of the error, which can then be corrected. This Demonstration allows you to simulate such a transmission by setting the data bits to be transmitted at the top, and introducing an error, if desired, in any position of the transmitted wor

Conway's M(13) puzzle

Educação Superior::Ciências Exatas e da Terra::MatemáticaJohn H. Conway introduced this sliding tile puzzle that bears the same relation to the Mathieu group M_12 as Sam Lloyd's famous 15 puzzle bears to the alternating group A_14. (M_12 was one of the first sporadic simple groups to be discovered.) Initially, tiles numbered 1–12 are placed on the points of a projective plane of order three, with the 13th point left uncovered. Hovering over a tile reveals the four points in the unique line of the projective plane that connects it to the uncovered point. Clicking the tile executes the sole legal move of the game, which is a double transposition: the clicked tile slides in to the uncovered space, and the other two points on the line exchange positions. The object, of course, is to restore the tiles to their initial order from a scrambled positio