fore no mechanism exists to detect errors. Figure 6.2 however shows all possible 4-bit words with valid odd-parity code- words in bold. As we can see there are eight valid 4-bit codewords as a subset of all 16 possible 4-bit sequences. The receiver can therefore detect whether errors have occurred by observing if the received sequence is a valid codeword. For example all valid codewords are shown in bold in Figure 6.2: if a codeword, such as '0011' is received, then a detection error has occurred. Similarly the repetition encoder of the example above limits the numbers of possible codes to only 2 out of 8 possible 3-bit sequences. Figure 6.3 shows all possible single bits, and Figure 6.4 all possible 3-bit codewords. The function of the encoder is to map the input bit sequence to a subset of all 3-bit sequences. The receiver is then designed to check the received {000,001,011,010,110,111,110,100} Figure 6.1 All possible 3-bit words 0000,0001,0011,0010,0110,0111,0101,0100,1100,1101,1111,1110,1010, 1011, 1001, 1000} Figure 6.2 All possible 4-bit words, with valid odd-parity codewords shown in bold {0,1} Figure 6.3 All possible 1-bit codewords {000,001,010,011,100,101,110,111} Figure 6.4 All possible 3-bit words, with the two valid codewords shown in bold Broadband Telecommunications Technologies and Management bit sequence against valid codewords and thereby decide whether an error has occurred and if SO try to correct the error using a maximum likelihood mechanism. As discussed, the above coding techniques are capable of detecting or correcting one error. Two or more errors will go either undetected or erroneously detected. More sophisticated cod- ing techniques have been developed that are capable of detecting/correcting many errors. All these techniques use mathematical processes based on modulo-2 binary arithmetic. Modulo-2 Operation The mathematical processes used in most coding schemes of interest to broadband telecom- munications are based on modulo-2 binary arithmetic, the rules of which are shown in Figure