Upload to Canvas a PDF of your work for the following problems.

Assignment: due Monday 12/5, 11:59 pm

  1. Use Macaulay2 for the following.

    1. Factor \(x^{15}+1\) over \(\mathbb{Z}_2\).
    2. For each factor \(f_i(x)\) that you found in part (a), factor \(f_i(x)\) over \(\text{GF}(16)\). Write each factor in the form \((x+\alpha^j)\), where \(\alpha\) is a primitive 15th root of unity.

  2. Find generators for \([15,5]\) and \([15,10]\) BCH codes as follows.

    1. Find a polynomial \(p(x)\) that generates a \([15,5]\) code in \(\mathbb{Z}_2/(x^{15} + 1)\) with designed distance 7. Show that this code has the required designed distance by showing the factorization of \(p(x)\) over \(\text{GF}(16)\).
    2. Find a polynomial \(p(x)\) that generates a \([15,10]\) code in \(\mathbb{Z}_2/(x^{15} + 1)\) with designed distance 4. Show that this code has the required designed distance by showing the factorization of \(p(x)\) over \(\text{GF}(16)\).

  3. The polynomial \(g(x) = x^4 + x^3 + 1\) generates a BCH code in \(\mathbb{Z}_2[x]/(x^{15}+1)\). Use the method given in the slides to correct the single error in the received word \(r(x) = x^{14}+x^{13}+x^{10}+x^9+x^7+x^5+x+1\). Give the error vector, along with the corrected code word.

  4. Let \(\alpha\) be a primitive 15th root of unity in \(\text{GF}(16)\). Find a generating polynomial for the distance 7 Reed-Solomon code in \(\text{GF}(16)[x]/(x^{15}+1)\). Write your polynomial as the sum of terms of the form \(\alpha^jx^k\), and give a generator matrix for the code whose nonzero entries all have the form \(\alpha^j\).

  5. Consider the code of Problem #4. Use the method given in the slides to correct the error in the following received word. \[r(x) = \alpha^{3}x^{14}+\alpha^{13}x^{13}+\alpha^{2}x^{12}+x^{11}+\alpha^{5}x^{10}+\alpha^{11}x^{9}+\alpha^{9}x^{8}+\alpha^{3}x^{6}+\alpha^{11}x^{5}+x^{4}+\alpha^{9}x^{3}+\alpha^{3}x^{2}+\alpha^{8}x+\alpha^{7}\] Give the error vector, along with the corrected code word.