Abstract Algebra: Theory and Applications

1.

Suppose a binary code has minimum distance $d=6\text{.}$ How many errors can be detected? How many errors can be corrected?

2.

Explain why it is impossible for the 8-bit string with decimal value ${56}_{10}$ to be an ASCII code for a character. Assume the leftmost bit of the string is being used as a parity-check bit.

3.

Suppose we receive the 8-bit string with decimal value ${56}_{10}$ when we are expecting ASCII characters with a parity-check bit in the first bit (leftmost). We know an error has occurred in transmission. Give one of the probable guesses for the character which was actually sent (other than ‘8’), under the assumption that any individual bit is rarely sent in error. Explain the logic of your answer. (You may need to consult a table of ASCII values online.)

4.

Suppose a linear code $C$ is created as the null space of the parity-check matrix

Then $x=11100$ is not a codeword. Describe a computation, and give the result of that computation, which verifies that $x$ is not a codeword of the code $C\text{.}$

5.

For $H$ and $x$ as in the previous question, suppose that $x$ is received as a message. Give a maximum likelihood decoding of the received message.