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

870 ERROR CORRECTING CODES
(a) For a single-error correcting (7, 4) systematic cyclic code with a generator polynomial
g (x) = x 3 + x 2 + I, find G and construct the code.
(b) Verify that this code is identical to that in Table 14.5 (Example 14.4).
14.3-8 (a) Use the generator polynomial g(x) = x 3 + x + 1 to find the generator matrix G' for a
nonsystematic (7, 4) cyclic code .
(b) Find the code generated by this matrix G 1 .
(c) Determine the error correcting capabilities of this code.
14.3-9 Use the generator polynomial g(x) = x 3 +x + 1 (see Prob. 14.3-8) to find the generator matrix
G for a systematic (7, 4) cyclic code.
14.3-10 Discuss the error correcting capabilities of an interleaved (>-.n, H) cyclic code with >-. = 10
and using a three-error correcting (31, 16) BCH code.
14.3-11 The generator polynomial
g(x) = x 10 +x 8 +x 5 +x 4 +x 2 +x+ 1
generates a cyclic BCH (15, 5) code.
(a) Determine the (cyclic) code generating matrix.
(b) For encoder input data d = 10110, find the corresponding codeword.
(c) Show how many errors this code can correct.
14.4-1 Uncoded data is transmitted by using PSK over an AWGN channel with E b! N = 9. This data
is now coded using a three-error correcting (23. 12) Golay code (Prob. 14.1-1) and transmitted
over the same channel at the same data rate and with the same transmitted power.
(a) Determine the corrected error probability Peu and Pee for the coded and the uncoded
systems.
(b) If it is decided to achieve the error probability P ee computed in part (a), using the uncoded
system by increasing the transmitted power, determine the required value of E b / N.
14.4-2 The simple code for detecting burst errors (Fig. 14.4) can also be used as a single-error correcting
code with a slight modification. The k data digits are divided into groups of b digits in length,
as in Fig. 14.4. To each group we add one parity check digit, so that each segment now has
b + 1 digits (b data digits and one parity check digit). The parity check digit is chosen to ensure
that the total number of ls in each segment of b + 1 digits is even. Now we consider these
digits as our new data and augment them with the last segment of b + 1 parity check digits, as
was done in Fig. 14.4. The data in Fig. 14.4 will be transmitted thus:
10111 01010 11011 10001 11000 01111
Show that this (30, 20) code is capable of single error correction as well as the detection of a
single burst of length 5.
14.5-1 For the convolutional encoder in Fig. 14.5, the received bits are 01 00 01 00 10 11
11 00. Use Viterbi's algorithm and the trellis diagram in Fig. 14.8 to decode this sequence.
14.5-2 For the convolutional encoder shown in Fig. P14.5-2:
(a) Draw the state and trellis diagrams and determine the output digit sequence for the data
digits 11010100.