1. Introduction

2. Literature review

3. General encoding and decoding algorithm

• Trellis Diagram

• Viterbi Algorithm

Figure 7. Received a codeword with two separated-bit error.

Figure 7. Received a codeword with two separated-bit error.

Figure 8. Received codeword with two adjacent errors.

Figure 8. Received codeword with two adjacent errors.

4. Comparative analysis Convolutional codes with LDPC codes, BCH codes and Turbo codes

4.2. Convolutional codes VS. BCH codes

BCH codes are a class of cyclic error-correcting codes that are constructed using polynomials over a finite field. They have precise control over the number of symbol errors correctable by the code and can be decoded easily using an algebraic method known as syndrome decoding. They are used in various applications such as satellite communications, DVDs, QR codes and quantum-resistant cryptography.

One way to compare the complexity of BCH codes and Convolutional codes is to look at their encoding and decoding algorithms.

In order to explicate the intricacies involved, we shall consider a (15, 7) BCH code and a (2, 1, 2) Convolutional code as illustrative examples. These codes have been chosen due to their comparable error correcting capabilities. This is solely a simplified illustration, and the actual complexity may vary depending on the implementation and the code parameters.

BCH codes: They are a class of cyclic error-correcting codes that are constructed using polynomials over a finite field. They have precise control over the number of symbol errors correctable by the code and can be decoded easily using an algebraic method known as syndrome decoding. They are used in various applications such as satellite communications, DVDs, QR codes and quantum-resistant cryptography.

BCH encoding: It is the process of finding a polynomial that is a multiple of the generator polynomial, which is defined as the least common multiple of the minimal polynomials of some consecutive powers of a primitive element. The encoding can be either systematic or non-systematic, depending on how the message is embedded in the codeword polynomial.

To encode a message using BCH code, need to choose a finite field GF(q), a code length n, and a design distance d. Then need to find a generator polynomial g(x) that has ∝, ${\propto }^2$,…,${\propto }^{(\mathrm{d}-1)}$ as roots, where ∝ is a primitive element of GF(${\mathrm{q}}^{\mathrm{m}}$) and n = ${\mathrm{q}}^{\mathrm{m}}$ - 1. Can be used the table of minimal polynomials in [28] to find g(x) as the least common multiple of some of them. Then multiply the message polynomial m(x) by g(x) to get the codeword polynomial c(x).

BCH decoding: It is the process of finding and correcting the errors in the received codeword using the syndromes, which are formed by evaluating the received polynomial at some powers of a primitive element. There are several algorithms for decoding BCH codes, such as Peterson’s algorithm, Forney algorithm, Sugiyama’s algorithm and Euclidean algorithm. They involve finding the error locator polynomial and the error values, and then subtracting them from the received codeword to recover the original one.

To decode a received word r(x) using BCH code, need to calculate the syndromes $s_{\mathrm{j}}$=r(${\propto }^{\mathrm{j}}$) for j = 1, 2, …, 2t, where t is the number of errors that can be corrected. Then use an algorithm such as Peterson’s algorithm or Euclidean algorithm to find the error locator polynomial $\wedge \left(x\right)$ that has the error locations as roots. Then use another algorithm such as Forney’s algorithm to find the error values at those locations. Finally, correct the errors by subtracting them from the received word to get the original codeword.

For example, suppose we want to encode the message [1 0 1 0] using a binary BCH code with n = 15 and d = 7, it is recommended to choose GF(2) as the finite field and use the reducing polynomial $z^4$+z+1 and the primitive element $\propto \left(z\right)$=z as in [28]. Then can find the generator polynomial g(x) as:

$$g\left(x\right)=\mathrm{lcm}(m_1\left(x\right),m_2\left(x\right),\cdots ,m_6\left(x\right))=(x^4+x+1)(x^4+x^3+x^2+x+1)=x^8+x^7+x^6+x^4+1$$

Please note that the lcm, which stands for least common multiple, is responsible for ensuring that g(x) contains all the necessary roots required for BCH encoding.

Now, The message polynomial is m(x) =$x^3$+x, so the codeword polynomial is

c(x) = m(x)g(x) = ($x^3$+x)($x^8$+$x^7$+$x^6$+$x^4$+1) =$x^{1}1$+$x^{1}0$+$x^9$+$x^7$+$x^5$+x

The codeword is [0 0 0 0 1 0 1 1 1 0 1 1 0 0 0].

Suppose receive the word r(x) = [0 0 0 0 1 0 1 0 1 0 1 1 0 0 0], which has two errors at positions $i_1$= 8 and $i_2$= 10. The syndromes can be computed as follows:

$s_1$= $r(\propto )$ = ∝$s_2$ = $r\left({\propto }^2\right)$ = ∝$s_3$ = $r\left({\propto }^3\right)$ = ∝$s_4$ = $r\left({\propto }^4\right)$ = ∝$s_5$ = $r\left({\propto }^5\right)$ = ∝$s_6$ = $r\left({\propto }^6\right)$ = ∝

Peterson’s algorithm can be used to find the error locator polynomial and as following:

$$\Lambda \left(x\right)=\mathrm{lcm}(m_{i_1}\left(x\right),m_{i_2}\left(x\right))=(x+{\alpha }^{15-i_1})(x+{\alpha }^{15-i_2})=(x+{\alpha }^7)(x+{\alpha }^5)=x^2+{\alpha }^{12}x+\alpha$$

while Forney’s algorithm can be employed to find the error values, and as follwoing:

$$e_{i_1}=\frac{s_1}{\Lambda ’\left({\alpha }^{15-i_1}\right)}=\frac{\alpha }{\left({\alpha }^{12-i_1}\right)}=\frac{\alpha }{\left({\alpha }^{12-8}\right)}=\frac{\alpha }{\left({\alpha }^4\right)}=\alpha$$

$$e_{i_2}=\frac{s_1}{\Lambda ’\left({\alpha }^{15-i_2}\right)}=\frac{\alpha }{\left({\alpha }^{12-i_2}\right)}=\frac{\alpha }{\left({\alpha }^{12-10}\right)}=\frac{\alpha }{\left({\alpha }^2\right)}=\alpha$$

To fix the errors, you can subtract them from the received word, and as following:

$$c\left(x\right)=r\left(x\right)-e\left(x\right)=\left[000010111011000\right]-\left[0000000\alpha \alpha \alpha \alpha 0000\right]=\left[000010100101000\right]$$

This is the same as the original codeword.

The complexity and the number of operations required for encoding and decoding BCH codes depend on the parameters of the code, such as the code length, the design distance, and the number of errors. Here are some general formulas for estimating the complexity:

For encoding, the complexity is proportional to the degree of the generator polynomial g(x), which is at most ($\mathrm{m}·\mathrm{t}$), where m is the extension degree of the finite field and t is the number of errors that can be corrected. The number of operations is O($\mathrm{m}·{\mathrm{t}}^2$) for binary BCH codes and O($\mathrm{m}·{\mathrm{t}}^2$ log q) for non-binary BCH codes, where q is the size of the finite field.

For decoding, the complexity is proportional to the number of syndromes that need to be computed and processed, which is ($2·\mathrm{t}$) for binary BCH codes and t for non-binary BCH codes. The number of operations is O(${\mathrm{t}}^2$ log n) for binary BCH codes and O(${\mathrm{t}}^3$ log q) for non-binary BCH codes, where N is the code length.

The given example is a binary BCH code because the finite field used is GF(2), which has only two elements (0 and 1).

Our objective entails the determination of the computational workload involved in both encoding and decoding a message within the provided example. The ultimate goal is to assess the inherent complexity embedded within the BCH code and subsequently conduct a comparative analysis between such complexity and the counterpart system comprised of Convolutional codes.

To know the number of encoding operation in BCH in the given example, we need to multiply the message polynomial m(x) by the generator polynomial g(x) to get the codeword polynomial c(x). The degree of g(x) is 8, which is equal to ($\mathrm{m}·\mathrm{t}$), where $\mathrm{m}=4$ and $\mathrm{t}=2$. The number of operations is O($\mathrm{m}·{\mathrm{t}}^2$) = O($4·2^2$) = $O\left(16\right)$ for binary BCH codes. This means that you need at most 16 operations in GF(2) to encode the message.

For decoding, need to calculate the syndromes, find the error locator polynomial, find the error values, and correct the errors. The number of operations depends on the number of errors and the algorithm used. For binary BCH codes, the number of operations is O($\mathrm{m}·{\mathrm{t}}^2$) for calculating the syndromes, O($\mathrm{m}·{\mathrm{t}}^3$) for finding the error locator polynomial using Peterson’s algorithm or O($\mathrm{m}·{\mathrm{t}}^2$) using Euclidean algorithm, O($\mathrm{m}·{\mathrm{t}}^2$) for finding the error values using Forney’s algorithm, and O($\mathrm{m}·\mathrm{t}$) for correcting the errors. The total number of operations is O($\mathrm{m}·{\mathrm{t}}^3$) for Peterson’s algorithm or O($\mathrm{m}·{\mathrm{t}}^2$) for Euclidean algorithm. In this example, $\mathrm{m}=4$ and $\mathrm{t}=2$, so the total number of operations is at most 128 for Peterson’s algorithm or 64 for Euclidean algorithm(note, this is the maximum number of operation, this means that the number of operation may be less than these numbers).

On the other hand, to encode the message [1 0 1 0] using a convolutional code (2, 1, 2), it is necessary to utilize the State Transition Table and the encoder circuit illustrated in Figure 3. Subsequently, the following operations need to be performed:

• Set all memory registers to zero.
• Feed the input bits one by one and shift the register values to the right.
• Calculate the output bits using the generator polynomials $g_1$ = (1,1,1) and $g_2$ = (1,0,1).
• Append two zero bits at the end of the message to flush the encoder.

For each input bit, we need to perform two additions and no multiplications. The total number of operations for encoding a message of length N plus m (m for the number of zeros that need to flush the shift registers), therefore the total number of operations for encoding a message of length N is 2N + m.

Figure 9, shows the encoding operation using Trellis diagram :

Figure 10, shows the decoding operation using Viterbi algorithm:

Figure 9. Message encoding using convolutional code (2,1,2).

Figure 9. Message encoding using convolutional code (2,1,2).

Figure 10. Codeword decoding operation using Viterbi Algorithm.

Figure 10. Codeword decoding operation using Viterbi Algorithm.

For encoding a message of length N = 4, we need to perform 2N+2 = 10 additions and no multiplications. For decoding a message of length N = 4+2 = 6 with four states in the trellis diagram, we need to perform approximately $N·4^2$=96 additions and 48 comparisons (there are 8 possible transitions per time step and 6 time steps for a message of length 4+2=6). Therefore, there are $8·6$=48 comparisons for these time steps (note, that the total of comparisons is less than 48, because the comparisons number in time step 1 and 2 is less than 8).

Another way to compare the complexity of BCH codes and Convolutional codes is to look at their codeword lengths. BCH codes tend to produce longer codewords than Convolutional codes for the same message length and error-correcting capability. This is because BCH codes have a fixed codeword length that depends on the size of the finite field, while Convolutional codes have a variable codeword length that depends on the constraint length and the code rate. For example, a (63,51) BCH code can correct up to 2 errors per codeword, while a (3,1) Convolutional code with constraint length 7 can correct up to 5 errors per codeword. However, the BCH code has a codeword length of 63 bits, while the Convolutional code has a codeword length of 21 bits for a message length of 7 bits [29,30].

When comparing the encoding and decoding operations, it is evident that Convolutional codes require fewer steps compared to BCH codes. Additionally, the mathematical operations involved in BCH codes are more complicated than in Convolutional codes. However, Convolutional codes still face a fundamental issue known as exponential complexity. Despite this drawback, Convolutional codes are considered the most appropriate choice for GPS communication systems.

Overall, the high complexity associated with BCH codes makes them unsuitable for implementation in GPS systems and real-world applications. Consequently, Convolutional codes with Viterbi decoding, which are less computationally complex, are preferred over BCH codes..

5. Conclusions

Abbreviations

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