nother valid codeword (in other words, an error occurs and is not detected.) This error probability is a function of the length of the input bit sequence and therefore is variable. 9CRC16(X) =x16+x2++x5+ 1 Figure 6.6 3G standards CRC polynomials of lengths 24, 16, 12 and 8 [2, 3] Superscript(2)Modulo-2 binary prime numbers are similar to decimal prime number in that their divisors are only themselves and 1. Error Control Coding Also, generally the larger the size of the CRC polynomial, the greater is its ability in detecting whether errors have occurred. Regardless, CRC techniques provide very good performance and their undetected error probability is very small. The reason that several CRC polynomials have been standardised is because different applications have different degrees of sensitivity to error events and therefore use an appropriate polynomial size. Case Study 6.1: Error Detection in Ethernet An experimental system of decentralised communication between computers was built and demonstrated by engineers at Xerox Palo Alto Research Center led by Robert Metcalf. They named it Ethernet, as they perceived its communication facility to be ether and shared among many computers. Ethernet was designed to allow computers to communicate with each other autonomously and without a centralised control. The medium connecting the computers was a coaxial cable and the equipment could operate at 3 megabits per second (Mbps) [4]. Each Ethernet packet was designed as shown in Figure 6.7. It began with a synchro- nisation sequence, followed by the addresses of the sender and the receiver, the data payload and ended with a checksum. An Ethernet packet is transmitted over a medium where transmission errors may occur, SO error control coding is necessary. The checksum is a CRC parity sequence of length 16 bits, which can detect possible reception errors with very high probability. The exact error detection probability is difficult to find as it depends on the distribution of errors. However one rule of thumb indicates that a 