Cryptology and Coding Theory
Group Homework #1
Due Thursday 14 September 2006
Adams Chapter 1 Problems: 1.2, 1.3, 1.5, 1.7, 1.11
Non-book:
1. Consider the binary repetition code of length 7.
a. How many errors can this code correct within one codeword? How many
errors can it detect? Carefully explain your answers.
b. Write down an expression that can be used to evaluate the probability of
decoding incorrectly when the message 0 is transmitted as 0000000, and
the probability of bit error is p. Explain your expression, but do not try to
evaluate your expression numerically!
2. Write a paragraph or two comparing and contrasting the binary repetition code
of length 5 with the Reed-Muller (AKA Hadamard) code. Discuss your findings
from the reading questions in Sect 1.3 as you see fit. (This should be a well-written
paragraph, proof-read and polished by all members of your group.)
3. Given the binary repetition code of length 5 consisting of codewords {00000,
11111}, find a 1 x 5 matrix G so that [0]G = [00000] and [1]G = [11111].
Notice that G is being used to encode the message 0 as the codeword 00000,
and similarly G encodes the message 1 as the codeword 11111.
4. Given the binary code C consisting of codewords {0000, 1100, 0011, 1111}, find
a 2 x 4 matrix G that can be used (as above) to encode 00 as 0000, encode 01
as 1100, encode 10 as 0011, and 11 as 1111.
Note: In each of 3) and 4) above, since the codes are binary, for each message m and
corresponding codeword c, the encoding process mG = c is performed using binary
arithmetic.
Please include a short report of how your group worked.