HW Number 2

Due on March 11th

  1. Find a quadratic polynomial \(f \in \mathbb{F}_7[x] \) such that $$ f \equiv x^5 + 3x^4 +5x + 1 \mod{(x^3 + x + 1)} $$
  2. Let \(G\) be the group formed by all rigid motion symmetries of an equilateral triangle (imagine a solid metal triangle with corners labelled A,B,C for instance).
  3. Let \(H = \{ (0, 0, 0, 0, 0, 0), (1, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 1, 1)\} \) be a subgroup of \(\mathbb{F}_2^6\) under vector addition modulo 2. Find all of the cosets of \(H\). Find the element of each coset with smallest Hamming weight.
  4. Find a basis of the nullspace of $$ \left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 \end{array}\right) $$ over \(\mathbb{F}_2\).