1.
True/False: There exists a perfect \([120,50,11]_2\) code.
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Worksheet Weekly Practice 5
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.
True/False: The Gilbert-Varshamov bound states that whenever
\begin{equation*}
M\cdot V_q(n,d-1) \lt q^n
\end{equation*}
then every code over \(GF(q)\) of length \(n\) and size \(M\) has minimum distance at least \(d\text{.}\)
3.
True/False: If \(V_q(n,d-1)\leq ((q-1)/2)q^{n-k}\) then more than half of the linear \([n,k]_q\) codes have minimum distance at least \(d\text{.}\)
Short Response.
Your responses to these questions should be complete solutions with justifications, as per the Grading Specifications.
4.
Using all of the bounds on the maximum possible size of a code of length \(n\) and distance \(d\) over \(GF(q)\) that we have learned so far (Hamming or Sphere-Packing Bound, Singleton Bound, Gilbert-Varshamov Bound for General Codes, Gilbert-Varshamov Bound for Linear Codes, The Plotkin Bound), tell me everything you can about the largest possible non-linear and linear codes of length \(10\) and distance \(5\) over \(GF(3)\text{.}\)
Solution.
We start by computing all of these values, displayed in the table below.
| Bound | Result |
|---|---|
| Hamming | \(M\leq 293\) |
| Singleton | \(k\leq 6, M\leq 729\) |
| Gilbert-Varshamov General | \(M\geq 59049/4521 \approx 13.06\) |
| Gilbert-Varshamov Linear | \(k \geq \lfloor \log_3( 59049/835)\rfloor = 3 \) |
| Plotkin | \(5 \leq \dfrac{10(2)3^{k-1}}{3^k -1}\) holds for any \(k\) |
| It is equivalent to \(0\leq 3^{k-1}+1\) |
From these we see that the largest possible general code of length \(10\) and distance \(5\) over \(GF(3)\) has between \(27\) (since there is a non-linear code of the same size as any linear code) and \(293\) elements, while the largest possible linear code has dimension between \(3\) and \(5\) (since the Hamming bound is more restrictive than the Singleton bound here).
