 6.5. Only addition and multiplication are shown here as subtraction is equal to addition and division can easily be derived from multiplication and addition. Example 6.3 What is the modulo-2 addition of '011' and '111'? What is the modulo-2 multiplication of '011' and '111'? Answer + 111 X 111 01100 01001 Codeword Distance Modulo-2 operation facilitates the measurement of an important metric in error control coding, the codeword distance, defined as the number of places two codewords differ. For example the following two odd parity bit codewords '11010101' and '01010111' differ in two places. Two codewords '000' and '111' differ in three places and therefore the code- word distance is 3. Codeword distance can be calculated using modulo-2 addition and counting the number of 1's of the sum. For example the modulo-2 addition of '11010101' Modulo-2 Addition: Modulo-2 Multiplication: 0 00 = 0 000 0 0 + 1 = 1 O X =0 1 0 0 = 1 1 0 0 = 0 + 1 = 0 Figure 6.5 Modulo-2 arithmetic rules Error Control Coding and '01010111' equals '10000010', and since the sum has two 1's the codeword distance is 2. Codeword distance is also known as the Hamming distance, named in honour of Richard Hamming. Codeword distance is an important measure of an error detection coding method as it shows the number of error events that can lead to false error detection. For example '11010101' may be transmitted and because of two error events be detected as '01010111'. However since the resulting codeword is valid, that is has an odd number of 1's, the error event goes undetected by the decoder. It follows that increasing the codeword distance decreases the probability that a decoding error may occur. However, increasing the code distance can mostly be achieved by increasing the number of parity bits, resulting in reduced code efficiency and increased cost. Appropriate code complexity and performance level are usually determined for different applications and specified within relevant standards. It follows that a coding technique's effectiveness in det