B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

822 ERROR CORRECTING CODES
Bose-Chaudhuri-Hocquenghen (BCH) Codes and
Reed-Solomon Codes
The BCH codes are perhaps the best studied class of random error correcting cyclic codes.
Moreover, their decoding procedure can be implemented simply. The Hamming code is a
special case of BCH codes. These codes are described as follows: for any positive integers m and
t (t < 2 m - I ), there exists at-error correcting (n, k) code with n = 2
m
- I and n - k :S mt. The
minimum distance dmin betweencodewords is bounded by the inequality 2t+l :S d m in :S 2t+2.
As a special case of nonbinary BCH codes, Reed-Solomon codes are by far the most
successful forward error correction (FEC) codes in practice today. Reed-Solomon codes have
found broad applications in digital storage (DVD, CD-ROM), high-speed modems, broadband
wireless systems, and HDTV, among numerous others. The detailed treatment of BCH codes
and Reed-Solomon codes requires extensive use of modem algebra and is beyond the scope
of this introductory chapter. For in-depth discussion of BCH codes and Reed-Solomon codes,
the reader is referred to the classic text by Lin and Costello. 2
Cyclic Redundancy Check (CRC) Codes for Error Detection
One of the most widely used cyclic codes is the cyclic redundancy check codes for detection
of data transmission errors. CRC codes are cyclic, designed to detect erroneous data packets
at the receivers ( often after error correction). To verify the integrity of the payload data block
(packet), each data packet is encoded by CRC codes of length n :S 2 m - I. The most common
CRC codes have m =12, 16, or 32 with code generator polynomial of the form
g (x) = (I + x)gc(x)
gc (x) = generator polynomial of a cyclic Hamming code
To select a code generator matrix, the design criterion is to control the probability of undetected
error events. In other words, the CRC codes must be able to detect the most likely error patterns
such that the probability of undetected errors
P (e H T = O le -::fa O) < E (14.26)
where E is set by the user according to its quality requirement. The most common CRC codes
are given in Table 14.7 along with their generator matrices. For each frame of data bits at
the transmitter, the CRC encoder generates a frame-checking sequence (FCS) of length 8, 12,
16, or 32 bits for error detection. For example, the IEEE 802.11 and IEEE 802. llb packets
are checked by the 16-bit sequence of CRC-CCITT, whereas the IEEE 802. lla packets are
checked by the CRC-32 sequence.
14.5 THE EFFECTS OF ERROR CORRECTION
Comparison of Coded and Uncoded Systems
It is instructive to compare the bit error probabilities (or bit error rate) when coded and uncoded
schemes are under similar constraints of power and information rate.
Let us consider a t-error correcting (n, k) code. In this case, k information digits are coded
into n digits. For a proper comparison, we shall assume that k information digits are transmitted
in the same time interval over both systems and that the transmitted power S; is also maintained