4
$\begingroup$

Let $p\geq 3$ be any prime and consider the code $C = N(H)\subseteq\mathbb{Z}_p^2$, where $H = \begin{pmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 0 & 1 & 2 & \dots & p-2 & p-1 \end{pmatrix}\in \mathbb{Z}_p^{2\times p}$.

a) How many codewords does the code $C$ contain? I've got an idea on this part. It should just be $p^2$, shouldn't it?

b) Show that every selection of two distinct columns of $H$ results in a non-singular 2x2 matrix.

c) Find a codeword in $C$ of weight 3 and use (b) to conclude that the code $C$ has distance 3.

Parts b and c I have no clue on where to start. I'm sure that I could probably reason my way through c if I could figure out b, since finding the codeword of weight 3 should be trivial. However, I'm willing to take any assistance on this problem.

  • 0
    What's $N(H)$? Is it the nullspace of $H$? If so, shouldn't it be in ${\bf Z}_2^p$, not ${\bf Z}_2^2$?2011-04-27
  • 0
    Yes, it should. My apologies.2011-04-27
  • 0
    $\mathbf{Z}_p^p$ would be even better notation for the vector space containing $C$ as a subspace :-)2011-06-22

3 Answers 3