The G can constructed from gx by first constructing the parity submatrix P as 1

The g can constructed from gx by first constructing

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Cyclic codes can be realized by the generating matrix, G. The G can constructed from g ( x ) by first constructing the parity submatrix, P, as 1. 1 st row of P: Rem ( x n – 1 / g ( x )) 2. 2 nd row of P: Rem ( x n – 2 / g ( x )) 3. : 4. k th row of P: Rem ( x n – k / g ( x ))
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ECE 550 Communication Theory – Channel Coding Hafiz Malik Example Decoding of Cyclic Codes For the cyclic code (7, 4) with , assume a received vector y = (1 1 0 0 0 0 0) 3 ( ) 1 g x x x Because s ( x ) = x , the syndrome vector is s =(0 1 0). This corresponds to the error pattern (0 0 0 0 0 1 0) from Table 15.2. The decoded codeword is therefore y = (1 1 0 0 0 0 0) + (0 0 0 0 0 1 0) = (1 1 0 0 0 1 0)
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ECE 550 Communication Theory – Channel Coding Hafiz Malik Cyclic Code Generation: Example Consider Hamming (7,4,3) code with generating polynomial g ( x ) = x 3 + x 2 + 1 Construct G and H matrixes
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ECE 550 Communication Theory – Channel Coding Hafiz Malik Example 15.5
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ECE 550 Communication Theory – Channel Coding Hafiz Malik Decoding of Cyclic Block Codes For decoding we exploit that fact that c ( x ) is devisable by g ( x ), if an errors during transmission, then the received polynomial r ( x ) will not be multiple of g ( x ). If the number of errors in r ( x ) are correctable then, r( x ) / g( x ) = m( x ) + s( x ) where syndrome polynomial s(x) = Rem (r(x) / g(x) ) has degree n – k – 1 or less. Let e ( x ) be the error polynomial, then, r(x) = c(x) + e(x) s(x) = Rem (e(x)/g(x))
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ECE 550 Communication Theory – Channel Coding Hafiz Malik BCH and Reed-Solomon Codes The BCH (Bose-Chaudhuri-Hocqunghen) codes is class of random cyclic codes. For m > 0 and t (t < 2 m -1 ), there exist a t-error correcting code (n, k) with n = 2 m -1, n – k <= mt, and 2t+1<= d min <= 2t + 2 Easy implementation makes them very popular. Hamming code is a special case of BCH codes. Reed-Solomon codes are a special case of nonbinary BCH codes Applications: DVD, CD-ROM, HDTV, high speed modems, boardband wireless systems, etc.
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ECE 550 Communication Theory – Channel Coding Hafiz Malik CRC Codes for Error Detection The CRC (Cyclic Redundancy Check) codes are used for error detection in the received data. The CRC codes are cyclic codes designed to detect errors at the reviewer. For integrity verification of the transmitted data packet, each packet at the transmitter is encoded by CRC codes of length n <= 2 m – 1 with code generating polynomial g ( x ) = (1 – x ) g c ( x ) where g c ( x ) is generator polynomial of a cyclic Hamming code The most commonly used m = 12, 16, 32 1 : 1 : 16 1 : 12 1 : 8 5 12 16 2 15 16 1 2 3 11 12 2 4 6 7 8 x x x CCITT CRC x x x CRC x x x x x CRC x x x x x CRC
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ECE 550 Communication Theory – Channel Coding Hafiz Malik Probability of Codeword Error: HD Decoding The probability of codeword error, P cw ( e ), is defined as the probability that a transmitted codeword is decoded in error Under hard-decision decoding, an ( n , k , d min ) block code can correct t or fewer errors if d min 2 t + 1 Thus a received vector may be decoded in error if it contains more than t errors. Therefore, P cw ( e ) can be expressed as By the application of union bound, we can write 1 ( ) 1 or more errors in codeword of length errors in codeword of length cw n j t P e P t n P j n  
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