A simple step-by-step decoding of binary BCH codes

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IEICE TRANS. FUNDAMENTALS, VOL.E88–A, NO.8 AUGUST 2005

LETTER

A Simple Step-by-Step Decoding of Binary BCH Codes

Ching-Lung CHR ^†a) , Szu-Lin SU ^† , Members, and Shao-Wei WU ^† , Nonmember

SUMMARY In this letter, we propose a simplified step-by-step decod- ing algorithm for t-error-correcting binary Bose-Chaudhuri- Hocquenghem (BCH) codes based on logical analysis. Compared to the conventional step- by-step decoding algorithm, the computation complexity of this decoder is much less, since it significantly reduces the matrix calculation and the op- erations of multiplication.

key words: BCH code, step-by-step decoding, matrix computation, com- putational complexity

1. Introduction

Among the most well-known error-correcting codes, the Bose-Chaudhuri-Hocquenghem (BCH) codes are a class of powerful random-error-correcting cyclic codes [1]–[3]. The popular binary BCH decoder architecture can be summa- rized into three steps: a) Calculate the syndromes from the received codeword. b) Compute the error locator polyno- mial. c) Find the error location, and then correct errors.

There is another decoding method, called the step-by- step decoding algorithm [4], can decode a cyclic code in a serial manner with a low hardware complexity. The method can directly determine whether any bit in received word is correct or not without finding the error-location polynomial.

Since the requirement for calculation of the determinant of the syndrome matrix, the conventional step-by-step decod- ing algorithm has not been widely employed for BCH codes with large error-correcting capability. To reduce the num- ber of matrix-calculations, [6] proposed a low-complexity step-by-step decoding algorithm for t-error-correcting bi- nary BCH codes. However, it is not the most simplified decoder.

This letter presents a novel step-by-step decoding al- gorithm for t-error-correcting binary BCH codes. Based on logical analysis, the determination whether a received bit is erroneous in the method as proposed in [6] can be further simplified into a simple equation. The novel decoder signif- icantly reduces the matrix calculations and the operations of multiplication compared with the algorithms in [5], [6].

2. Decoding Algorithm

An (n, k) t-error-correcting binary BCH code of block length Manuscript received November 17, 2004.

Manuscript revised March 9, 2005.

Final manuscript received May 6, 2005.

†

The authors are with the Department of Electrical Engineer- ing, National Cheng Kung University, Tainan 701, Taiwan, R.O.C.

a) E-mail: [email protected] DOI: 10.1093/ietfec/e88–a.8.2236

n = 2

− 1 can be defined in terms of the roots of its gen- erator polynomial. Let α be a primitive element in the Ga- lois field GF(2

), where m is an integer with m
3. Then the generator polynomial g(x) of the code is the lowest de- gree polynomial over GF(2), which has α, α ² , α ³ , · · · , α ^2t as its roots. Let Φ 1 (x) , Φ 3 (x) , · · · , Φ 2t−1 (x) be the distinct min- imum polynomials of α, α ³ , · · · , α ^2t−1 , respectively. Then, g(x) is given by g(x) = LCM [Φ 1 (x) , Φ 3 (x) , · · · , Φ 2t−1 (x)].

Now let v(x), e(x) and r(x) be a systematic codeword, the error polynomial and the received polynomial, respectively, and so these polynomials related by the expression r(x) = v(x) + e(x). The weight of the error pattern e(x) would be the number of errors in the received codeword r(x). The syndromes can be calculated by S

= r(α

) = e(α

), for i = 1, 2, · · · , 2t.

The principle of the conventional step-by-step decod- ing algorithm is that it involves changing the received bits one at a time by testing to determine whether the number of errors is reduced. For 1  v  t, the v × v syndrome matrix is defined as

L _v =

 







S ₁ 1 0 · · · 0

S 3 S 2 S 1 · · · 0 ... ... ... ... ...

S 2 v−1 S 2 v−2 S 2 v−3 · · · S _v

 







. (1)

The relationship between the syndrome matrices and the number of errors in r(x) can be given by theorem 9.11 in [1]. The theorem is rewritten as follows:

Theorem 1: For any binary BCH code and any v such that 1  v  t, the v×v syndrome matrix L _v is singular if the num- ber of errors is v-1 or less, and is nonsingular if the number of errors is v or v+1.

Using the theorem, the number of errors in r(x) can be determined from the values of the determinants of the syndrome matrices det(L _v ), v = 1, 2, · · · , t. For instance, if det(L 1 )
0, det(L 2 )
0, and det(L v ) = 0, for v = 3, 4, · · · , t, then two errors have occurred. Thus, in order to acquire the number of errors, the step-by-step decoding algorithm is concerned whether the values of det(L _v ) , v = 1, 2, · · · , t, equal to zero, and a decision vector l composed of decision bits l _v , is defined as [5]

l = (l 1 , l 2 , · · · , l

), (2)

where l _v = 0 if det(L v ) = 0 and l v = 1 if det(L v )
0.

Copyright c
2005 The Institute of Electronics, Information and Communication Engineers

LETTER

2237 Table 1 The possible various cases of e

for t = 3.

If the received word r(x) is modified by changing tem- porarily a selected bit at position x

, 0  p  n − 1, then the modified decision vector l ¯p can also be defined as

l _¯p = (l ¯1 , l ¯2 , · · · , l ¯t ) , (3) where the subscript “ ¯p” in l ¯p indicates that the magnitude of the x

position of the error pattern e(x) is temporarily changed. Whether the bit at position x

of r(x) is erroneous can be determined from the difference between l and l ¯p .

Using logical analysis, a novel step-by-step decoding algorithm for t-error-correcting binary BCH codes is pro- posed as follows. The logical analysis is based on the fact that only 2t+1 cases are possible when we want to deter- mine the value at the x

position of the error pattern e, e = (e 0 , e 1 , · · · , e

−1 ), for a t-error-correcting binary BCH code. For example, for a binary (15, 5) BCH code (t = 3), all seven possible cases of e

are expressed in Table 1 and described as below:

1. If no error occurs, then the value of e

must be correct, i.e., e

= 0.

2. If one error occurs, then the value of e

can be erro- neous or correct, i.e., e

= 1 or e

= 0.

3. If two errors occur, then the value of e

can be erro- neous or correct, i.e., e

= 1 or e

= 0.

4. If three errors occur, then the value of e

can be erro- neous or correct, i.e., e

= 1 or e

= 0.

Theorem 2: For a t-error-correcting binary BCH code, if v (1  v  t) errors occur, then the estimated error value at position p (0  p  n − 1) of the error pattern can be given by

ˆe

(t) = ¯l v, ¯p , f or 1  v  t (4)

Proof:

Case t = 1: Let v be the number of errors, and e

(1) be the value at the x

position of the error pattern. There are three possible cases apply in determining the value of e

(1), as shown in row 2 of Table 2. v ¯p and l _{1, ¯p} indicates the weight of the error pattern and the decision bit for changing the re- ceived digit r

, respectively. Row 4 is the decision bit l _{1, ¯p} . If no error occurs (v ¯p = 0), det(L 1 , ¯p = 0), then l 1 , ¯p = 0. If one error occurs ( v ¯p = 1), det(L 1, ¯p
0), then l 1, ¯p = 1. If two errors occur (v ¯p = 2), det(L 1 , ¯p
0), then l 1 , ¯p = 1. It is easy to see that ˆe

(1) = ¯l 1, ¯p .

Case t
2: Only 2t+1 possible cases when we determine the value of position x

of the error pattern, as shown in row 2 of Table 3. As the case of t = 1, the decision bits, l 2 , ¯p , l 3 , ¯p , · · · , and l

, ¯p in Table 3 can be determined by The- orem 1. It is easy to see that all 2t+1 possible estimated

Table 2 The logic analysis in Theorem 2 for t = 1.

Table 3 The logic analysis in Theorem 2 for t ≥ 2.

values of e

(t), as shown in row 7, 10 and 12 of Table 3, are equal to ¯l _{v, ¯p} . Hence, in a similar way, we can get

ˆe

(t) = ¯l _{v, ¯p} , for 2  v  t

Q.E.D.

The matrix calculations and the operations of multipli- cation needed for the algorithms in [5], [6] and the proposed algorithm are given in Table 4. Since the probability of a large number of errors is small, only a low-scale matrix cal- culation is needed in the decoder. Therefore, the proposed algorithm significantly reduces the matrix calculations and the operations of multiplication.

According to Theorem 2, the novel step-by-step decod- ing procedure can be presented as follows:

1. Determine the number of errors v. Detect consecu- tively det(L

), det(L

−1 ), · · · , until a nonzero determi- nant is found [1]. Then the number of errors is v.

2. Let p = n - 1.

3. Change the magnitude at position p of r(x) temporar- ily and determine the modified syndromes S

, ¯p (i = 1 , 2, · · · , 2t; n − k  p  n − 1).

4. Estimate the error value ˆe

(t) = ¯l _{v, ¯p} . 5. Send the output bit ˆr

= r

+ ˆe

(t).

6. Let p = p-1. If p = n-k-1 or all v errors have been found,

then this decoding algorithm is completed. Otherwise,

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IEICE TRANS. FUNDAMENTALS, VOL.E88–A, NO.8 AUGUST 2005

Table 4 The matrix calculations and multiplications of the estimated er- ror value for t = 3.

go to step 3.

3. Decoder Architecture

Based on the above step-by-step decoding algorithm, a hard- ware structure of the decoder is presented. Figure 1 shows the functional block diagram of the decoder. It consists of four parts:

1. n-stage buffer register: The buffer register is used to store the received vector and to correct received vector bit by bit.

2. Syndrome generator: The syndrome generator is used to obtain the original syndromes S

(i = 1, 2, 3, · · · , 2t) and the temporarily changed syndromes S

, ¯p (i = 1, 2, 3, · · · , 2t; n − k  p  n − 1).

3. Error-number calculator: The error-number calculator is used to calculate the determinants det(L _v ), 1  v  t, and then determine the number of errors v.

Fig. 1 Functional block diagram of the binary step by step decoder.

4. Error-corrector: The error-corrector is used to estimate the error value ˆe

(t) = ¯l v, ¯p . If the corresponding bit is judged to be an erroneous bit, the decoder sends a correcting bit ˆe

(t) = 1 to change its magnitude.

4. Example of Decoding of Binary (15, 5) BCH Code Let α be a primitive element of GF(2 ⁴ ), satisfying α ⁴ + α + 1 = 0. The generator polynomial of the binary (15, 5) BCH code is defined as the least common multiple of the minimal polynomials of 1, α, α ² , · · · , and α ⁶ . Assume c(x) = 0 and e(x) = x ³ + x ¹⁴ are the codeword and the error polynomial, respectively. There are two errors in the received polyno- mial r(x). The initial syndromes are

S ₁ = 1, S 2 = 1, S 3 = α ⁸ , S 4 = 1, S 5 = α ⁵ , S 5 = α ¹ . Detect consecutively det(L

):

Since

det(L 3 ) = 

 

S 1 1 0

S ₃ S ₂ S ₁ S 5 S 4 S 3

 

 = 0 (5)

and

det(L 2 ) = 

 S 1 1 S ₃ S ₂

  = α ²
0, (6) so l ₃ = 0 and l 2 = 1.

Then, we confirm that two errors (v = 2) occur and the esti- mated error value of e

(3) is

ˆe

(3) = ¯l 2 , ¯p .

For 10  p  14, the temporarily changed syndrome values can be given by

S

, ¯p = S

+ α

.

Then,

LETTER

2239

S _{1, ¯ 14} = α ³ , S _{1, ¯} 13 = α ⁶ , S _{1, ¯} 12 = α ¹¹ , S _{1, ¯} 11 = α ¹² , S _{1, ¯} 10 = α ⁵ , S 2 , ¯ 14 = α ⁶ , S 2 , ¯ 13 = α ¹² , S 2 , ¯ 12 = α ⁷ , S 2 , ¯ 11 = α ⁹ , S 2 , ¯ 10 = α ¹⁰ , S _{3, ¯} 14 = α ⁹ , S _{3, ¯} 13 = α ¹² , S _{3, ¯} 12 = α ¹⁴ , S _{3, ¯} 11 = α ¹³ , S _{3, ¯} 10 = α ² .

By calculating the value of det(L _{2, ¯p} ) = 

 S _{1, ¯p} 1 S 3 , ¯p S 2 , ¯p

 , (7)

we can determine

l _{2, ¯ 14} = 0, l _{2, ¯} 13 = 1, l _{2, ¯} 12 = 1, l _{2, ¯} 11 = 1, l _{2, ¯} 10 = 1, and

ˆe 14 = ¯l _{2, ¯} 14 = 1, ˆe 13 = ¯l _{2, ¯} 13 = 0, ˆe 12 = ¯l _{2, ¯} 12 = 0, ˆe 11 = ¯l _{2, ¯} 11 = 0, ˆe 10 = ¯l _{2, ¯} 10 = 0.

Hence, the estimated error values of the message part are (ˆe 10 , ˆe 11 , ˆe 12 , ˆe 13 , ˆe 14 ) = (0, 0, 0, 0, 1).

5. Conclusions

A modified step-by-step decoding algorithm for t-error- correcting binary BCH codes has been presented. Based on

logical analysis, we obtain a simple rule for estimating the value of the error pattern. The proposed algorithm signifi- cantly reduces the matrix calculations and the operations of multiplication as the algorithms in [5], [6]. Thus, the com- putational complexity of this decoder is much less, since the most complex element of the step-by-step decoder is the

“matrix-computing” element. Furthermore, the simple and regular structure also makes it suitable for hardware imple- mentation.

References

[1] W.W. Peterson and E.J. Weldon, Error-Correcting Codes, MIT Press, Cambridge, MA, 1972.

[2] E.R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, NY, 1968.

[3] S. Lin and D.J. Costello, Error Control Coding: Fundamentals and Applications, Prentice-Hall, Englewood Cli ffs, NJ, 1983.

[4] J.L. Massey, “Step-by-step decoding of the Bose-Chaudhuri- Hocquenghem codes,” IEEE Trans. Inf. Theory, vol.IT-11, no.4, pp.580–585, 1965.

[5] S.W. Wei and C.H. Wei, “A high-speed real-time binary BCH de- coder,” IEEE Trans. Circuits Syst. Video Technol., vol.3, no.2, pp.138–147, 1993.

[6] C.-L. Chr, S.-L. Su, and S.-W. Wu, “A low-complexiy step-by-step

decoding algorithm for binary BCH codes,” IEICE Trans. Fundamen-

tals, vol.E88-A, no.1, pp.359–365, Jan. 2005.