1.
Fill-In: Fill in the missing part of the statement of Proposition 2.3.7.
For every \([n,k,d]_q\) code \(C\) with parity check matrix \(H\text{,}\) the minimum distance \(d\) of \(C\) is equal to .
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Worksheet Weekly Practice 3
Instructions: You may type up or handwrite your work, but it must be neat, professional, and organized and it must be saved as a PDF file and uploaded to the appropriate Gradescope assignment. Use a scanner or scanning app to convert handwritten work on paper to PDF. I encourage you to type your work using the provided template.
All tasks below must have a complete solution that represents a good-faith attempt at being right to receive engagement credits. If your submission is complete and turned in on time, you will receive full engagement credit for the assignment. All other submissions will receive zero engagement credit. Read the guidelines at Grading Specifications carefully.
To abide by the class academic honesty policy, your work should represent your own understanding in your own words. If you work with other students, you must clearly indicate who you worked with in your submission. The same is true for using tools like generative AI although I strongly discourage you from using such tools since you need to build your own understanding here to do well on exams.
True/False, Multiple Choice, & Fill-In.
For these problems a justification is not required for credit, but it may be useful for your own understanding to include one. True/False problems should be marked True if the statement is always true, and False otherwise. Multiple choice problems may have more than one correct answer if that is indicated in the problem statement; be sure to select all that apply. Fill-in problems require a short answer such as a number, word, or phrase.
2.
Fill-In: The number of elements in a vector space of dimension \(k\) over the finite field \(GF(q)\) is \(\fillinmath{q^k} \)
3.
True/False: The binary Hamming code \(C_{H,r}\) has parameters \([2^r - 1, 2^r - r - 1, 3]_2\text{.}\)
4.
Multiple Choice: Which statement below about decoding a linear 1-error-correcting code with generator matrix \(G\) and parity-check matrix \(H\) is true?
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To decode a received word \(\mathbf{y}\text{,}\) we must check the distance from \(\mathbf{y}\) to every codeword.
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If \(\mathbf{y}\) is the received word, then no errors have occurred if \(H\transpose{\mathbf{y}} \neq \mathbf{0}\text{.}\)
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If \(\mathbf{y}\) is the received word, then no errors have occurred if \(\mathbf{y}G =\mathbf{0}\text{.}\)
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If \(\mathbf{y}\) is the received word, then \(H\transpose{\mathbf{y}}\) is a multiple of a column of \(H\) if one error has occurred.
Short Response.
Your responses to these questions should be complete solutions with justifications, as per the Grading Specifications.
5.
Determine the parameters \([n,k,d]_q\) of the code with parity-check matrix over \(GF(3)\)
\begin{equation*}
H= \begin{bmatrix}
1 \amp 0 \amp 1 \amp 1 \amp 2 \amp 0 \\
0 \amp 1 \amp 2 \amp 2 \amp 1 \amp 0 \\
0 \amp 0 \amp 1 \amp 0 \amp 2 \amp 2
\end{bmatrix}\text{.}
\end{equation*}
6.
7.
Use the efficient decoding algorithm for 1-error-correcting codes to decode the received words \((1111111)\) and \((1110011)\) using the binary Hamming code \(C_{H,3}\text{.}\)
