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

868 ERROR CORRECTING CODES
(c) Determine the error correcting capabilities of this code.
(d) Prepare a suitable decoding table.
(e) Decode the following received words: 101100, 000110, 101010.
14.2-10 (a) Construct a code table for the (6, 3) code generated by the matrix G in Prob. 14.2-6.
(b) Prepare a suitable decoding table.
14.2-11 Construct a single-error correcting (7, 4) linear block code (Hamming code) and the
corresponding decoding table.
14.2-12 For the (6, 3) code in Example 14.1, the decoding table is Table 14.3. Show that if we use this
decoding table, and a two-error pattern 010100 or 001001 occurs, it will not be corrected. If
it is desired to correct a single two-error pattern 010100 (along with six single-error patterns),
construct the appropriate decoding table and verify that it does indeed correct one two-e1rnr
pattern 010100 and that it cannot correct any other two-error patterns.
14.2-13 (a) Given k = 8, find the minimum value of n for a code that can c01rect at least one error.
(b) Choose a generator matrix G for this code.
(c) How many double errors can this code correct?
(d) Construct a decoding table (syndromes and corresponding correctable error patterns).
14.2-14 Consider a (6, 2) code generated by the matrix
[ 0 1
0 ]
(a) Construct the code table for this code and determine the minimum distance between
codewords.
(b) Prepare a suitable decoding table.
Hint: This code can correct all single-error patterns, seven double-error patterns, and two
triple-error patterns. Choose the desired seven double-error patterns and the two triple-error
patterns.
14.3-1 (a) Use the generator polynomial g(x) = x 3 + x + Ito construct a systematic (7, 4) cyclic
code.
(b) What are the error correcting capabilities of this code?
(c) Construct the decoding table.
(d) If the received word is 1101100, determine the transmitted data word.
14.3-2 A three-error correcting (23, 12) Golay code is a cyclic code with a generator polynomial
g(x) = x ll +x 9 +x 7 +x 6 +x 5 +x+ 1
Determine the codewords for the data vectors 000011110000, 101010101010, and
11000101011110.
14.3-3 Factorize the polynomial