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Introduction to Reed-Solomon Coding ( Part I )

Introduction to Reed-Solomon Coding ( Part I ). Introduction. One of the most error control codes is Reed-Solomon codes. These codes were developed by Reed & Solomon in June, 1960.

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Introduction to Reed-Solomon Coding ( Part I )

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

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

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

  4. Effect of Noise

  5. digital data 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 Reconstructed data 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 encoder 0 0 0 0 check bits, r=2 1 1 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 encoder 000 111 000 111 111 000 000 receiver 000 101 000 111 111 010 001  decoder 000 111 000 111 111 000 000

  6. A (7,4) HAMMING CODE 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

  7. Let a1, a2, ..., ak be the k binary of message digital. Let c1, c2, ..., cr be the r parity check bits. An n-digital codeword can be given by a1a2a3...akc1c2c3...cr n bits The check bits are chosen to satisfy the r=n-k equations, 0 = h11a1h12a2...h1kakc1 0 = h21a1h22a2...h2kakc2 . . (1) . 0 = hr1a1hr2a2...hrkakcr

  8. Equation (1) can be writen in matrix notation, h11 h12 ... h1k 1 0 ... 0 a1 0 h21 h22 ... h2k 0 1 ... 0 a2 0 . . . . ak = 0 . c1 0 . c2 0 . . . hr1 hr2 ... hrk 0 0 ... 1 cr 0 rn n1 r1  H  T = 0

  9. Let E be an n1 error pattern at least one error, that is e1 0 e2 0 . . E = . = ej = 1 . . . . en 0 Also let R be the received codeword, that is r1 a1 0 r2 a2 0 . . . R = . = T + E = ak + ej = 1 . c1 . . . . rn cr 0

  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 s1 h11 h12 0 s2 h21 h22 0 . = . e1 + . e2 + ... + . en (2) . . . . . . . . sr hr1 hr2 1

  11. Assume e1=0, e2=1, e3=0, ..., en=0 s1 h12 s2 h22 . = . . . . . sr hr2 The syndrome is equal to the second column of the parity check matrix H. Thus, the second position of received codeword is error.

  12. A (n,k) hamming code has n=r+k=2r-1, where k is message bits and r=n-k is parity check bits. • The rate of the hamming code is given by • Hamming code is a single error correcting code. • 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.

  13. Single-error-correcting Binary BCH code In GF(24), let p(x)=x4+x+1 be a primitive irreducible polynomial over GF(24). Then the elements of GF(24) are

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

  15. Decoder: • Let received word be R = C + E codeword error pattern H‧R=H(C+E)=H‧C+H‧ET=H‧ET = where • Let R=C+E=(11100101001001)+(00100....0) =(11000101001001)

  16. Let Information polynomial be I(x)= • The codeword is Information polynomial parity check polynomial R(x) I(x)

  17. 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). • For example, to encode a (15,11) BCH code, the generator polynomial is g(x)=x4+x+1, where α is a order of 15 in GF(24) and is called a minimum polynomial of α.

  18. To encode, one needs to find C3,C2,C1,C0 or R(x) = such that satisfies • To show this, dividing I(x) by g(x), one obtains I(x)=Q(x)g(x)+R(x) • Encoder C(x)=I(x)+R(x)=Q(x)*g(x) • Since C(x) is a multiple of g(x); then C(x)=I(x)+R(x) is a (15,11) BCH code.

  19. Example : I(x)=Q(x)g(x)+R(x) C(x)=Q(x)g(x)=I(x)+R(x) = =111001010011001 …

  20. To decode, let the error polynomial is E(x)= • The received word polynomial is R’(x)=C(x)+E(x)= • The syndrome is = = is the error location in a received word.

  21. Double-error-correcting Binary BCH code • To encode a (n=15, I=7) BCH code over GF(24), which can correct two errors. • Let C(x)=K(x)g1(x)g2(x) where g1(α) is the minimal polynomial of α. => g1(α) = 0 g2(α3) is the minimal polynomial of α3. => g2(α3) = 0

  22. The minimal polynomial of αis • The minimal polynomial of α3 is • The generator polynomial of a(15,7) BCH code is

  23. Reed-Solomon (RS) code • An RS code is a cyclic symbol error-correcting code. • 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. • 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.

  24. In order to be able to correct ‘t’ symbol errors, the minimum distance of the code words ‘D’ is given by D=2t+1. • 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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