1.
True/False: When defining a GRS code over \(\GF(q)\text{,}\) any choice of \(n\) evaluation points \(\alpha_1,\alpha_2,\ldots,\alpha_n\) and nonzero weights \(v_1,v_2,\ldots,v_n\) and dimension \(k\) will yield a code with the same parameters.
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Worksheet Weekly Practice 9
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: When defining a RS code over \(\GF(q)\text{,}\) we can choose any length \(n\leq q-1\text{.}\)
Short Response.
Your responses to these questions should be complete solutions with justifications, as per the Grading Specifications. The polynomial computation tools from SageMath may be helpful once you understand how to do the computations.
3.
Construct the generator polynomial \(g(x)\) for the narrow-sense primitive Reed-Solomon code over \(F=\GF(16)=\F_2[\alpha]/(\alpha^4+\alpha+1)\) with length \(n=15\) and distance \(d=7\) and generator element \(\alpha\text{.}\)
4.
Construct the generator matrix for the encoding of the code from the previous problem using the encoding map \(\mathcal{E}\) that sends \(m(x)\mapsto m(x)g(x)\) and records the coefficent sequence of the resulting polynomial as \((c_0\, c_1\, \dots\, c_{n-1})\text{.}\)
Note: if you are using LaTeX to write your solution and are using the to your preamble to allow for more than 10 columns in the matrix.
bmatrix or pmatrix environment, you will need to add \setcounter{MaxMatrixCols}{15}
5.
6.
Encode the messages \(m_1(x)\) and \(m_2(x)\) using the encoding map \(\mathcal{E}^*\) that sends \(m(x)\mapsto x^{n-k}m(x)-r_m(x)\) where \(r_m(x)\) is the remainder on division of \(x^{n-k}m(x)\) by \(g(x)\) and records the coefficent sequence of the resulting polynomial as \((c_0\, c_1\, \dots\, c_{n-1})\text{.}\)
