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Figure 2.3 Standard array for an n; k linear code.
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LINEAR CODES
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Coset leaders v0 000000 e1 : 100000 e2 : 010000 e3 : 001000 e4 : 000100 e5 : 000010 e6 : 000001 e7 : 100001
Syndromes v1 100110 000110 110110 101110 100010 100100 100111 000111 v2 010101 110101 000101 011101 010001 101111 010100 110100 v3 001011 101011 011011 000011 001111 001001 001010 101010 v4 110011 010011 100011 111011 110111 110001 110010 010010 v5 101101 001101 111101 100101 101001 101111 101100 001100 v6 011110 111110 001110 010110 011010 011100 011111 111111 v7 111000 011000 101000 110000 111100 111010 111001 011001 000 110 101 011 100 010 001 111
Figure 2.4 Standard array for (6, 3) code.
2n =2k 2n k 2r distinct rows in the standard array, and each row consists of 2k distinct words. These 2r rows are called cosets of the code C and the rst n-tuple ej of each coset is called a coset leader, mentioned in Subsection 2.1.1. Example 2.5 We consider the 6; 3 linear code expressed by the following H matrix: 1 H 41 0 2 1 0 1 0 1 1 0 1 0 0 1 0 3 0 0 5: 1
The standard array of this code is shown in Figure 2.4. A standard array of an n; k linear code C consists of 2k disjoint columns, meaning 2k disjoint subsets. Each column consists of 2n k 2r n-tuple words, with the topmost one as a codeword in C. Let Dj denote the j-th column of the standard array. Then Dj fvj e 1 vj e 2 vj ... e2n k 1 vj g; 2:5
where vj is a codeword of C and e0 v0 ; e1 ; e2 ; . . . ; e2n k 1 , are the coset leaders. The 2k distinct columns D0 , D1 ; . . . ; D2k 1 , can be used for decoding the code C. Assume that the codeword vj is transmitted. From Eq. (2.5) the received word r is in Dj if the error pattern is a coset leader. In this case the received word r is decoded correctly into the transmitted codeword vj . On the other hand, if the error pattern is not a coset leader, an erroneous decoding will be performed. That is, the decoding is correct if and only if the error pattern is a coset leader. For this reason the 2n k 2r coset leaders including the zero codeword are called the correctable error patterns. Every word included in the same coset has the same syndrome, and in addition no two syndromes in different cosets are equal. Therefore every n; k linear block code is capable of correcting 2n k error patterns. The decoding the received word in the standard array is performed in the following steps: Step 1. Compute the syndrome of r, that is, r HT . Step 2. Locate the coset leader ei whose syndrome is equal to r HT . Then ei is assumed to be the error pattern. Step 3. Decode the received word r into the codeword v r ei .
MATHEMATICAL BACKGROUND AND MATRIX CODES
BASIC MATRIX CODES
In this section the typical error control codes are introduced. These codes are basic to the design of practical codes that should t the requirements of future applications. As was mentioned before, a linear code can be expressed by a parity-check matrix or by a generator polynomial. Here we will use matrix form of expression for the basic codes. 2.3.1 Simple Parity-Check Codes
In digital systems parity check is usually used to detect errors because this requires only one check bit and is implemented by very simple encoder / decoder hardware. Parity check has been extensively applied to logic systems and memory systems, including data-path logic circuits, arithmetic logic circuits, high-speed memories, and so forth. Among the variety of error control codes the simple parity-check code is the easiest to use, and it is rst presented precisely to help the reader understand the matrix code. A parity-check bit is determined to make the total number of 1 s in a codeword even. For example, assume that an eight-bit input word d 0 1 1 1 0 1 0 1 is given. Since d includes ve 1 s, a parity bit p is determined to be 1 in order to make the total number of 1 s even. The parity bit p is appended to d, and this results in the codeword v d p 0 1 1 1 0 1 0 1 1 having even number of 1 s. For this reason a simple parity-check code is sometimes called an even parity code. Alternatively, this encoding procedure can also be performed for odd number of 1 s; then it is called an odd parity code. The above encoding procedure can be expressed in mathematical form. That is, for a given k-bit input word d d0 d1 . . . dk 1 , where di 2 GF 2 , 0 i k 1, a parity bit p is generated by p d HT ; e d0 d1 dk 1 ; where He 1 1 . . . 1 with the row vector consisting of k 1 s. The calculation by addition is performed over GF 2 , so the denotes modulo-2 addition. The codeword of the simple parity-check code C is v d p d0 d1 . . . dk 1 p . The decoding procedure is as follows: for the received word r with k 1 bits, meaning 0 0 0 r d0 p0 d0 d1 . . . dk 1 p0 , a parity check is performed as S r HT
0 0 0 d0 d1 dk 1 p0 1 1 1 1 T 0 0 0 d0 d1 dk 1 p0 ;
where H is a row vector with k 1 1 s, or H 1 1 1 1 : | {z }
2:6
Calculation is then performed over GF 2 , and S is the parity-checked output, called a syndrome. If an error exists in the received word, the error e e0 ; e1 ; . . . ; ek 1 ; ep ,