Contents Introduction Primitive BCH Codes Generator Polynomial Properties Decoding of BCH Codes Syndrome Computation Syndrome and Error Pattern

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Conrad Carr
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1 BCH CODE

2 Contents. Introduction. Primitive BCH Codes. Generator Polynomial 4. Properties 5. Decoding of BCH Codes 6. yndrome Computation 7. yndrome and Error Pattern 8. Error-location Polynomial 9. Decoding Procedure for BCH Codes

3 ContentsCont. 0. Berlekamp s Iterative Method for Finding σ. Finding the Roots of σ. The tep-by-tep Decoding

4 . Introduction BCH Bose Chaudhuri Hocquenghem codes form a large class of multiple random error-correcting codes. They were first discovered by Hocquenghem in 959, and independently by Bose and Chaudhuri in 960. They are cyclic codes. They are constructed based on Galois fields. 4

5 . Primitive BCH Codes For any integers m and t < m-, there exists a primitive BCH code with the following parameters: n m n k mt d min t This code is capable of correcting t or fewer random error over a span of m bit positions. The parameter t is called the designed errorcorrecting capability and the parameter t is called the designed minimum distance. 5

6 For example, for m 6 and t, there exists a BCH code with n 6-6 n k 6 8 d min 7 The code is a triple-error-correcting 6, 45 BCH code. 6

7 . Generator Polynomial Let be a primitive element in Galois field GF m. For i t, let φ i- be the minimum polynomial of the field element i-. The degree of φ i- is m or a factor of m. The generator polynomial g of a t-errorcorrecting primitive BCH code of length m is given by g LCM{ φ, φ, Lφt } 7

8 g Note that the degree of is mt or less. Hence the number of parity-check bits, n k, of the code is at most mt. Example : Let m 4 and t. Let be a primitive element in GF 4 which is constructed based on the 4 primitive polynomial p. The minimum polynomials of, and 5 are: φ 4 φ 4 φ 5 8

9 9 Hence the generator polynomial of the triple-errorcorrecting BCH code of length n 4 5 is The code is a 5, 5 cyclic code },, { LCM g φ φ φ φ φ φ

10 Example : Let m 6 and t 5. Let be a primitive element in GF 6 which is constructed based on the 6 primitive polynomial p. The minimum polynomials of,, 5, 7 and 9 are: φ 6 φ 4 6 φ φ 7 6 φ 9 Hence the generator polynomial of the 5-errorcorrecting BCH code of length n 6 6 is 0

11 g LCM{ φ, φ, φ5, φ7, φ9 φ φ φ φ φ } The degree of g is 7. Consequently, the code is a 6, 6 cyclic code.

12 Table : A list of primitive BCH codes

13 Table : A list of primitive BCH codes Cont.

14 Table : The elements of GF 4 generated by p 4 4

15 Table : Minimal polynomials of the elements in GF 4 generated by p 4 cont. Conjugate roots Minimal polynomials 0,, 4, 8 4, 6, 9, 4 5, 0 7,,, 4 4 5

16 Table : Minimal polynomials for GF m For example, the minimal polynomial of is φ, which is denoted by 0,, The conjugate roots of φ are, 6, 6 5 6

17 Table : Minimal polynomials for GF m cont. 76

18 Table : Minimal polynomials for GF m cont. 87

19 Table : Minimal polynomials for GF m cont. 98

20 Table : Minimal polynomials for GF m cont. 0 9

21 Table : Minimal polynomials for GF m cont. 0

22 4. Properties Consider a t-error-correcting BCH code of length n m with generator polynomial g g has,,,, t as root, i.e., i g 0, for i t. ince a code polynomial is a multiple of V g V also has,,,, t as roots, i.e., i V 0, for i t.

23 A polynomial V of degree less than m is a code polynomial if and only if it has,,,, t as roots.

24 5. Decoding of BCH Codes Consider a t-error-correcting BCH code of length n m with generator polynomial uppose a code polynomial V V v 0 v v... v n n is transmitted. r r 0 r r... Let be the received polynomial. r n n 4

25 Then r V e, where e is the error pattern caused by the channel noise. To check whether r is a code polynomial or not, we simply test whether. r r... r If yes, then r is a code polynomial; otherwise r is not a code polynomial and the presence of errors is detected. Decoding of a BCH code consists of the same three steps as for the decoding a cyclic code, namely: syndrome computation, determination of the error pattern, and error correction. t 0 5

26 6. yndrome Computation The syndrome consists of t components in GF m,,,..., t, i where r for i t. Computation: Let φ i be the minimum polynomial of i. Dividing r by φ i, we obtain Then i r a φi b i i b i i b can be obtained by feedback shift-register with connections based on φ i. 6

27 Example : Let m 4 and t. Consider the doubleerror correcting BCH code of length 4 5. The generator polynomial has,,, 4, as roots where is a primitive element in GF 4 4 constructed based on p. The code is a 5, 7 code. uppose the vector, is received. Then r r 8 7

28 The minimum polynomial of,,, 4 is φ φ φ 4 4. The minimum polynomial of is φ 4. Dividing r by φ and φ, we have b b 8

29 Then b b 4 4 b b Hence the syndrome of r is,,, 4 7 8,,, 4 9

30 Example 4: this example uses the built functions in MATLAB6. to simulate a Chinese poem transmission over the AWGN channel as NR 6.0 db. clear fid fopen' 杜甫詩.txt','r'; A freadfid; % A is an array of integers chara'; % to chinese character D sizea; NR 6 ; % 6dB NR 0logEb/No0logsignal_pw/code_rate**nois e_var, assume signal_pw noise_var /*code_rate*0^nr/0; 0

31 KK 6; % info length NN ; % code length TT ; % error correct capability for i::d for j:8 % to get all bits in two adjacent integers bj bitgetai,j; bj8 bitgetai,j; end u bchencob,nn,kk; % bch,6, encoder v -*u; % with BPK format, 0 <-->, <--> - r v sqrtnoise_var*randnsizev; % AWGN channel for j: if rj > 0 yj 0; % hard decision output else yj ;

32 end end y y6:; % without decoded, obtain the first 8-bit. y y4:; % without decoded, obtain the second 8-bit. Bi bitsnumy,8; Bi bitsnumy,8; v_hat bchdecoy,kk,tt; Ci bitsnumv_hat:8,8; Ci bitsnumv_hat9:6,8 ; end fid fopen' 杜甫詩雜訊干擾.txt','w'; fid fopen' 杜甫詩 BCH decoded.txt','w'; fprintffid,'%c',charb; fprintffid,'%c',charc; fclose'all';

33 烽火連三月家書抵萬金白頭騷更短渾欲不勝簪烽火連三月頡股抵萬? 白貄騷更短渾欲提細穠? 烽火連三月家書抵萬金白頭騷更短渾欲不勝簪國破山河在城春草木深感時花濺淚恨別鳥驚心國? 廾河在城春秦 d 麮 c 感時兕濺? 鳥驚心國破山河在城春草木深感時花濺淚恨別鳥驚心 Figure : the original Chinese poem left, degraded by AWGN middle, recovered with BCH. encoder/decoder right

34 HW #9 g. What is the generator polynomial of the three-error-correcting of BCH code with length, which employed in Example 4?. Referring to Example 4, please input a new Chinese poem, and what is the result of this poem with BPK signal transmission over the Rayleigh fading channel as NR 6.0?. 4

35 7. yndrome and Error Pattern ince, then for i t. r v e i i i i r v e i e, This gives a relationship between the syndrome and the error pattern. uppose e has v errors at the locations j j j,,..., v, i.e., where e j j L 0 j < j <... < j < n. v v j, 5

36 6 From and, we have the following relation between the syndrome components and error locations : t j t j t j t t j j j j j j v v v e e e L M L L

37 If we can solve these t equations, we can j j j v determine,, L,. j v Once, j, L, j are determined, the exponents j, j,, j v tell us the locations of errors. Any method for solving these t equations is a decoding method. j v ince the elements, j, L, j give the locations of errors, they are called error-location numbers. For simplicity, let βl j l, for l v 7

38 Then, we have β β... β β β... β v β β... β t t t t v v M 4 These equations are known as power-sum symmetric functions. 8

39 8. Error-location Polynomial Define σ β β... β 0 σ σ Lσ where σ 0. σ has β, β, L, βv the reciprocals of errorlocation numbers as roots. σ is called the error-location polynomial. If we can determine σ from the syndrome, then the roots of σ give us the error location numbers, and hence the error pattern can be determined. v v v 5 9

40 From 5, we have the following relationship between the coefficients of σ and the errorlocation numbers: σ 0 σ β β Lβ σ β β β β Lβ β v σ βββ βββ Lβ β β 4 v v v σ β β Lβ v M v v v 6 40

41 The above equations are called elementary symmetric functions. From 4 and 6, we have the following relationship between the syndrome and the coefficients of σ: σ 0 σ σ 0 σ σ σ 0 M σ σ σ Lσ vσ v v v v v v σ σ σ Lσ v v v v v M

42 Note that 0. Then iσ σ, ; { 0, i for odd i i for even i The equations of 7 are called the Newton s identities. If we can determine σ, σ,, σ v from the Newton s identities, then we can determine the error-location numbers, β, β,, β v by finding the roots of σ. 4

43 A procedure for finding the error pattern yndrome Error-location Polynomial σ Error-location Numbers Error Patterns e 4

44 9. Decoding Procedure for BCH Codes Compute the syndrome Find error location polynomial σ. Determine the error-location numbers by finding the roots of σ. 4 Correct errors 44

45 Example 5: As we mentioned in Example, if the allzero vector is transmitted, i.e., V V and the received vector is, r r 8 The syndrome is computed as,,, ,,, 45

46 46 From the Newton s identities 7, we obtain the following equations ince it is a -error-correcting BCH code, the errorlocation polynomial is σ σ σ σ σ σ σ σ σ σ 0 σ σ σ σ 0 4 σ σ

47 47 We substitutes these syndrome values and the Newton s identities become o that, σ σ σ σ σ σ 8, σ σ > 8 σ

48 48 The error location polynomial is factored as The roots are and. Their multiplication inverse elements is and. Therefore the error location numbers are and. The error polynomial is The decoded polynomial is β β σ β e 0 ˆ V e r V 8 β

49 0. Berlekamp s Iterative Method for Finding σ σ can be computed iteratively. The iteration process consists of t steps. At the u-th step, we determine a minimum-degree polynomial lu σ σ σ Lσ u u u u l whose coefficients satisfy the first u Newton s identities u 49

50 Our next step is to find σ u whose coefficients satisfy the first u Newton s identities. First we check whether σ u also satisfies the u -th Newton s identity. If yes, σ u σ u is a minimum degree polynomial whose coefficients satisfy the first u Newton identities. If not, a correction term is added to σ u to form σ u so that its coefficients satisfy the first u Newton s identities. 50

51 To test whether σ u satisfies the u -th Newton s identity, we computes u u u du u σ u σ u... σl u u lu This quantity is called the u-th discrepancy. If d u 0, then the coefficients of σ u satisfies the u -th Newton s identity. We set σ u σ u l u l u actually, l u is the degree of σ u If d u 0, σ u needs to be adjusted to satisfy the u -th Newton s identity. 5

52 Correction: We go back to the steps prior to the u- th step and determine a polynomial σ p such that d and p - l p has the largest value, where p 0 l p is the degree of σ p. Then σ σ d d σ u u u p p u p σ u is the solution at the u -th step of the iteration process. Repeat the testing and correction until we reach the t-th step. Then σ σ t. The above iteration method applies to both binary and nonbinary BCH codes. 5

53 For binary BCH codes, it can be reduced to t steps. Every even step can be skipped. Execution of the Iteration Process Note that σ t satisfies the first Newton s identity. To carry out the iteration, we set up a table as below and fill out the table: u σ u d u l u u - l u M t 5

54 . Finding the Roots of σ The roots of σ in GF m in σ. If σ i 0, then i is a root of σ and σ -i m i is an error-location number. Roots error-location numbers determination and error correction can be carried out simultaneously. To decode the first received digit r n-, we check whether is a root of σ. If σ 0, then r n- is erroneous and must be corrected. If σ 0, then r n- is error-free. 54

55 Iterative formula, u, t σ σ d d σ u u u p p u p d u u u u u σ u σ u Lσ l u u l u u u u du u σ u σ u... σl u u lu 55

56 To decode r n-i, we test whether σ i 0. If σ i 0, r n-i is erroneous and must be corrected, otherwise r n-i is error-free. Example 6: Consider the decoding of the 5, 5 triple-error-correcting BCH code given in Example. The generator polynomial has,,, 4, 5, 6 as roots. The roots, and 4 have the same minimum polynomial, φ φ φ

57 The roots and 6 have the same minimum polynomial, φ φ 6 4 The minimum polynomial of 5 is uppose V is transmitted and is received. r φ

58 58 Then Clearly the error pattern is Dividing by φ, φ and φ 5 respectively, we have the following remainders: 5 r 5 e 5 b b b

59 59 The syndrome components are: Hence b b b b b b,,,,, 5 0 0

60 The st step: d σ 0 The nd step: σ d σ σ The rd step: since d 0 and in comparison of the values of d u and u l in steps and 0. u p 0 is taken i.e. step 0,

61 u σ σ d d σ u u u p p u p d u u u u u σ u σ u Lσ l u u l u σ σ d d 0 0 σ 0 5 d 0 4 σ 0 σ 5 6

62 d d σ σ σ σ σ d The 4 th step: 4 σ σ σ σ d

63 The 5 th step: consider the values of du and u l prior to the 4 th step. We take p i.e. nd step u σ 5 σ d 5 6 σ 5 5 σ 5 4 σ The 6 th step: σ σ 6

64 Iterative process results in the following table : u σ u d u l u u - l u take p take p

65 Iterative process results in the following table : Note that σ σ 6 5 σ σ σ 0 0 Hence, 0 and are roots of σ. The reciprocals of these roots are -, -0 5 and -. Hence, 5 and are the error-location numbers. Consequently, the error pattern is e 5 65

66 HW #0. In Example 6, what are the error pattern e and the output of BCH decoder, if the received vector is r

67 67. The tep-by-tep Decoding In this decoding, we do not find the error-location polynomial. Instead, we use the concept of the error-trapping decoding. First we define the syndrome matrix as following: where v v v v v M L M M L L ,,, t L

68 Theorem : For any binary BCH n, k, t code, and any v such that v t, the v by v syndrome matrix is singular if the number of errors is v- or less, and is nonsingular if the number of errors is v or v. The decision vector is defined m m, m, L, m t where decision bit is calculated as m v m m v v 0 if if det M det M v v

69 The decision vector of a general t-error-correcting binary BCH code can be determined as follows: if there are no errors, then m 0,0, L,0 if there is one error, then m,0, L,0 if there are u errors, then m {,,,0 0 where the symbol can be 0 or. 4if there are no less than t errors, the t,0 u t u t } m { u,,} 69

70 For example, if t, the decision vector could be 0, 0 for no errors,, 0 for single error, and, for two errors. For a received sequence r r0, r, Lrn, and error pattern with weight L, the step-by-step decoding is et i t, j - Check if detm i 0, If detm i 0, i i-, go to step 4. j j, Complement r n-j- to determine the modified syndrome and whether detm i 0. If detm i 0, set r n j rn j.otherwise, i i-. If i 0, go to 5. 70

71 4If j < k, go to. If j k, go to 5. 5Read the information digits r n-, r n-,., r n-k Example 7: in Example, for -error-correcting 5, 7 BCH code, suppose the all-zero vector is transmitted. and the received sequence is r The syndrome is computed as follows: r 9 7

72 r r 0 det det M There are at least two errors at the received sequence. To complement, the received sequence polynomial becomes as follows: r 4

73 7 4 9 r The syndrome is computed as follows: r r 0 det det M From the syndrome matrix, it shows that to complement does not reduce the number of errors. r 4

74 74 For p > 0, is obtained by cyclically shifting p places to the right. The syndrome matrix for is defined as follows: p r p v p v p v p v p p p p p v M L M L L p r r

75 The modified step-by-step decoding is described as follows: et i t, p. 0 0 Check det M i 0, if det M i 0, i i-, go to step 4. p Complement r to determine the shift 0 p syndrome and whether det M i 0. p If det M, set r p. 0 r p i 0 0 Otherwise, r n p r n p, and i i-. If i 0, go to 5. 4 p p, if p < k, go to. If p k, stop. 5Read the information digits r n-, r n-,., r n-k 75

76 Example 8: Consider -error-correcting 5, 7 BCH code over GF 4 with the primitive polynomial 4. The generator polynomial is g φ φ 8 uppose the all-zero vector is transmitted. And the received sequence is r The whole procedure of the step-by-step decoding is shown in the following page. After 7 steps, the information vector is obtained. 76

77 77 9 r r r det det det 0 0 M M

78 The corresponding decision vector is m, As one time cycle shift in r, the corresponding syndrome is r r

79 det M det M det 8 8 det 8 The corresponding decision vector is m, The whole decoding process is shown as follows. 79

80 p r m r p p 8 4,,, m 8,,,, p p 4 p4 p , Info. r ,,,, ,,,, ,,, 7,,, p ,,,, p ,,,, , ,

81 Example 8: This example uses the built functions in MATLAB 6. to simulate a photo transmission over the AWGN channel as NR 6.0 db. All statements in this program are almost as Example 4. The original photo, deteriorated photo and recovered photo are shown in the following page. Example 9: this example uses the built functions in MATLAB 6. to simulate a sound record transmission over the AWGN channel as NR 6.0 db. These three program files are attached on website 老胡小舖 8

82 Figure : the original photo 8

83 Figure : the deteriorated photo by AWGN channel. 8

84 Figure 4: the recovered photo with BCH encoder/decoder 84

85 HW #0-. Referring to Example 6, what is the result of this photo with BPK signal transmission over the Rayleigh fading channel as NR 6.0? Compare with the results of Example 6.. Referring to Example 7, what is the result of this sound record with BPK signal transmission over the Rayleigh fading channel as NR 6.0? Compare with the results of Example 7.. Referring to Example, find the BCH decoded sequence. 85

Chapter 6. BCH Codes

Chapter 6. BCH Codes Chapter 6 BCH Codes Description of the Codes Decoding of the BCH Codes Outline Implementation of Galois Field Arithmetic Implementation of Error Correction Nonbinary BCH Codes and Reed-Solomon Codes Weight