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L. J. Wang 1 Introduction to Reed-Solomon Coding ( Part I )

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L. J. Wang 2 Introduction o One of the most error control codes is Reed- Solomon codes. o These codes were developed by Reed & Solomon in June, 1960. o The paper I.S. Reed and Gus Solomon, “ Polynominal codes over certain finite fields ”, Journal of the society for industrial & applied mathematics.

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L. J. Wang 3 o Reed-Solomon (RS) codes have many applications such as compact disc (CD, VCD, DVD), deep space exploration, HDTV, computer memory, and spread-spectrum systems. o In the decades, since RS discovery, RS codes are the most frequency used digital error control codes in the world.

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L. J. Wang 4 Effect of Noise

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L. J. Wang 5 digital data0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 Reconstructed data0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 encoder0 0 0 0 check bits, r=2 1 1 information bits, k=1 block length of code, n=3 000, 111 code word a ( n, k ) code, n=3, k=1, and r=n-k=3-1=2 code rate, p=k/n=1/3 encoder000 111 000 111 111 000 000 receiver000 101 000 111 111 010 001 decoder000 111 000 111 111 000 000

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L. J. Wang 6 A (7,4) hamming code n=7, k=4, r=n-k=7-4=3, p=4/7. 0101 1100 1001 0000 I1 I2 I3 I4 encoder receiver decoder A (7,4) HAMMING CODE

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L. J. Wang 7 Let a 1, a 2,..., a k be the k binary of message digital. Let c 1, c 2,..., c r be the r parity check bits. An n-digital codeword can be given by a 1 a 2 a 3...a k c 1 c 2 c 3...c r n bits The check bits are chosen to satisfy the r=n-k equations, 0 = h 11 a 1 h 12 a 2 ... h 1k a k c 1 0 = h 21 a 1 h 22 a 2 ... h 2k a k c 2..(1). 0 = h r1 a 1 h r2 a 2 ... h rk a k c r

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L. J. Wang 8 Equation (1) can be writen in matrix notation, h 11 h 12... h 1k 1 0... 0 a 1 0 h 21 h 22... h 2k 0 1... 0 a 2 0.... ak=0. c10. c20... h r1 h r2... h rk 0 0... 1 c r 0 r n n 1 r 1 H T = 0

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L. J. Wang 9 Let E be an n 1 error pattern at least one error, that is e 1 0 e 2 0.. E =. =e j = 1.. e n 0 Also let R be the received codeword, that is r 1 a 1 0 r 2 a 2 0... R =.= T + E =a k +e j = 1.c 1.... r n c r 0

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L. J. Wang 10 Thus S = H R = H (T+E) = H T + H E = H E S = H E where S is an r 1 syndorme pattern. Problem, for given S, Find E s 1 h 11 h 12 0 s 2 h 21 h 22 0. =. e 1 +. e 2 +... +. e n (2).... s r h r1 h r2 1

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L. J. Wang 11 Assume e 1 =0, e 2 =1, e 3 =0,..., e n =0 s 1 h 12 s 2 h 22. =... s r h r2 The syndrome is equal to the second column of the parity check matrix H. Thus, the second position of received codeword is error.

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L. J. Wang 12 o A (n,k) hamming code has n=r+k=2 r -1, where k is message bits and r=n-k is parity check bits. o The rate of the hamming code is given by o Hamming code is a single error correcting code. o In order to correct two or more errors, cyclic binary code, BCH code and Reed-Solomon code are developed to correct t errors, where t ≧ 1.

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L. J. Wang 13 In GF(2 4 ), let p(x)=x 4 +x+1 be a primitive irreducible polynomial over GF(2 4 ). Then the elements of GF(2 4 ) are Single-error-correcting Binary BCH code

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L. J. Wang 14 o The parity check matrix of a (n=15,k=11) BCH code for correcting one error is o Encoder: o Let the codeword of this code is information bitsparity check bits

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L. J. Wang 15

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L. J. Wang 16 o Decoder: o Let received word be R= C+E codeworderror pattern H ‧ R=H(C+E)=H ‧ C+H ‧ E T =H ‧ E T = where o Let R=C+E=(11100101001001)+(00100....0) =(11000101001001)

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L. J. Wang 17

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L. J. Wang 18 o Let Information polynomial be I(x)= o The codeword is Information polynomial parity check polynomial I(x) R(x)

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L. J. Wang 19 o Note that C(x)=Q(x) ‧ g(x) where g(x) is called a generator polynomial, C(x) is a codeword if and only if C(x) is a multiple of g(x). o For example, to encode a (15,11) BCH code, the generator polynomial is g(x)=x 4 +x+1, where α is a order of 15 in GF(2 4 ) and is called a minimum polynomial of α.

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L. J. Wang 20 o To encode, one needs to find C 3,C 2,C 1,C 0 or R(x) = such that satisfies o To show this, dividing I(x) by g(x), one obtains I(x)=Q(x)g(x)+R(x) o Encoder C(x)=I(x)+R(x)=Q(x)*g(x) o Since C(x) is a multiple of g(x); then C(x)=I(x)+R(x) is a (15,11) BCH code.

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L. J. Wang 21 o Example : I(x)=Q(x)g(x)+R(x) C(x)=Q(x)g(x)=I(x)+R(x) = =111001010011001 …

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L. J. Wang 22 o To decode, let the error polynomial is E(x)= o The received word polynomial is R’(x)=C(x)+E(x)= o The syndrome is = is the error location in a received word.

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L. J. Wang 23 o To encode a (n=15, I=7) BCH code over GF(2 4 ), which can correct two errors. o Let C(x)=K(x)g 1 (x)g 2 (x) where g 1 (α) is the minimal polynomial of α. => g 1 (α) = 0 g 2 (α 3 ) is the minimal polynomial of α 3. => g 2 (α 3 ) = 0 Double-error-correcting Binary BCH code

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L. J. Wang 24 o The minimal polynomial of αis o The minimal polynomial of α 3 is o The generator polynomial of a(15,7) BCH code is

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L. J. Wang 25 o An RS code is a cyclic symbol error-correcting code. o An RS codeword will consist of I information or message symbols, together with P parity or check symbols. The word length is N=I+P. o The symbols in an RS codeword are usually not binary, i.e., each symbol is represent by more than one bit. In fact, a favorite choice is to use 8-bit symbols. This is related to the fact that most computers have word length of 8 bits or multiples of 8 bits. Reed-Solomon (RS) code

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L. J. Wang 26 o In order to be able to correct ‘t’ symbol errors, the minimum distance of the code words ‘D’ is given by D=2t+1. o If the minimum distance of an RS code is D, and the word length is N, then the number of message symbols I in a word is given by I = N – ( D – 1 ) P = D – 1.

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Error Control Code. Widely used in many areas, like communications, DVD, data storage… In communications, because of noise, you can never be sure that.

Error Control Code. Widely used in many areas, like communications, DVD, data storage… In communications, because of noise, you can never be sure that.

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