BCH Codes

National Instruments has partnered with Prentice Hall to bring you large portions of in-depth technical topics from several PTR RF and Communications books, including Digital Communications: Fundamentals and Applications, 2nd Edition. This series of content is designed for a broad range of audiences, from experts who want to review a specific topic to students who need easy-to-understand documentation for their projects.

For the complete list of RF topics, please visit the RF and Communications Resource Main Page.

6.8.3 BCH Codes

Bose–Chadhuri–Hocquenghem (BCH) codes are a generalization of Hamming

codes that allow multiple error correction. They are a powerful class of cyclic codes

that provide a large selection of block lengths, code rates, alphabet sizes, and errorcorrecting

capability. Table 6.4 lists some code generators g(x) commonly used for

the construction of BCH codes [8] for various values of n, k, and t, up to a block

length of 255. The coefficients of g(x) are presented as octal numbers arranged so

that when they are converted to binary digits the rightmost digit corresponds to the

zero-degree coefficient of g(x). From Table 6.4, one can easily verify a cyclic code

property—the generator polynomial is of degree n − k. BCH codes are important

because at block lengths of a few hundred, the BCH codes outperform all other

block codes with the same block length and code rate. The most commonly used

BCH codes employ a binary alphabet and a codeword block length of n = 2^m − 1,

where m = 3, 4, . . . .

The title of Table 6.4 indicates that the generators shown are for those BCH

codes known as primitive codes. The term “primitive” is a number-theoretic concept

requiring an algebraic development [7, 10–11], which is presented in Section

8.1.4. In Figures 6.21 and 6.22 are plotted error performance curves of two BCH

codes (127, 64) and (127, 36), to illustrate comparative performance. Assuming

hard decision decoding, the PB versus channel error probability is shown in Figure

6.21. The P_B versus E_b/N₀ for coherently demodulated BPSK over a Gaussian

channel is shown in Figure 6.22. The curves in Figure 6.22 seem to depart from our

expectations. They each have the same block size, yet the more redundant (127, 36)

code does not exhibit as much coding gain as does the less redundant (127, 64)

code. It has been shown that a relatively broad maximum of coding gain versus

code rate for fixed n occurs roughly between coding rates of 1/3 and 3/4 for BCH codes

[12]. Performance over a Gaussian channel degrades substantially at very high or

very low rates [11].

Figure 6.23 represents computed performance of BCH codes [13] using coherently

demodulated BPSK with both hard- and soft-decision decoding. Soft-decision

decoding is not usually used with block codes because of its complexity. However,

whenever it is implemented, it offers an approximate 2-dB coding gain over harddecision

decoding. For a given code rate, the decoded error probability is known

to improve with increasing block length n [4]. Thus, for a given code rate, it is

Figure 6.23  P_B versus E_b/N₀ for coherently demodulated BPSK over a Gaussian channel

using BCH codes. (Reprinted with permission

from L. J. Weng, “Soft and Hard Decoding Performance Comparisons for BCH Codes,”

Proc. Int. Conf. Commun., 1979, Fig. 3, p. 25.5.5. © 1979 IEEE.)

interesting to consider the block length that would be required for the harddecision-

decoding performance to be comparable to the soft-decision-decoding performance.

In Figure 6.23, the BCH codes shown all have code rates of approximately 1/2.

From the figure [13] it appears that for a fixed code rate, the hard-decision-decoded

BCH code of length 8 times n or longer has a better performance (for P_B of about 10^-6 or less)

than that of a soft-decision-decoded BCH code of length n. One special subclass

of the BCH codes (the discovery of which preceded the BCH codes) is the particularly

useful nonbinary set called Reed-Solomon codes. They are described in Section 8.1.

Relevant NI products

Customers interested in this topic were also interested in the following NI products:

• RF and Communication Hardware and Software
  
• Other Modular Instruments (digital multimeters, digitizers, switching, etc...)
• LabVIEW Graphical Programming Environment

For the complete list of tutorials, return to the NI RF and Communications Fundamentals Main page.

Buy the Book

Purchase Digital Communications from Prentice Hall Professional through this link and receive the following:

• Between 15% and 30% Off
• Free Shipping and Handling

0 ratings | 0.00 out of 5

Print

Reader Comments | Submit a comment »