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TABLE 11.1
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Even Parity Codewords Modulo-2 addition of dataword 0 1 1 0 1 0 0 1 Codeword 0000 0011 0101 0110 1001 1010 1100 1111
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Dataword 000 001 010 011 100 101 110 111
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The Hamming distance between two codewords is defined as the number of positions by which the two codewords differ. Thus the codewords 0000 and 1111 differ in four positions, so their Hamming distance is four. The minimum Hamming distance, usually just referred to as the minimum distance is the smallest Hamming distance between any two codewords. It can be shown that the minimum distance is given by the minimum number of binary 1s in any codeword, excluding the all-zero codeword. By inspection it will be seen that the minimum distance of the code in Table 11.1 is two. The greater the minimum distance the better the code, as this reduces the chances of one codeword being converted to another by noise. The properties of linear block codes are best formulated in terms of matrices. Only a summary of some of these results are presented here, as background to aid in the understanding of coding methods used in satellite communications. A dataword (or message block) of size k is denoted by a row vector d, for example the sixth dataword in Table 11.1 is d6 [101]. Denoting the codeword by row vector c, the corresponding codeword is c6 [1010]. In general, the codeword is generated from the dataword by use of a generator matrix denoted by G, where c dG (11.2)
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Design of the generator matrix forms part of coding practice and will not be gone into here. However an example will illustrate the properties. One example of a generator matrix for a (7, 4) code is 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1
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(11.3)
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It will be noted that the matrix has 7 columns and 4 rows corresponding to the (7, 4) code, and furthermore, the first four columns form an identity submatrix. The identity submatrix results in the dataword appearing as the first four bits of the codeword, in this example. In general, a systematic code contains a sequence that is the dataword, and the most common arrangement is to have the dataword at the start of the codeword as shown in the example. It can be shown that any linear block code can be put into systematic form. The remaining bits in any row of G are responsible for generating the parity bits from the data bits. As an example, suppose it is required to generate a codeword for a dataword [1010]. This is done by multiplying d by G 1 0 0] 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1
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[1 0 1 0 0 1 0] The dataword is seen to appear as the first four bits in the codeword, and the end three bits are the parity bits. The parity bits are generated from the data bits by means of the last three columns in the generator matrix. This submatrix is denoted by P:
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1 1 P5 1 0
1 1 0 1
1 0 1 1
(11.4)
The transpose of P, which enters into the decoding process is formed by interchanging rows with columns, that is, row 1 becomes column 1, and column 1 becomes row 1, row 2 becomes column 2 and column 2 becomes row 2, and so on. In full, the transpose of P, written as PT is 1 PT 5 C 1 1 1 1 0 1 0 1 0 1S 1
What is termed the parity check matrix (denoted by H) is now formed by appending an identity matrix to PT: 1 H 5 C1 1 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 0S 1
(11.5)
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