Find all the lines in $\mathbb{F}_3^2$ (think back to Monday’s class and tic tac toe); draw them all and give them their equations as names.
If a line goes through the origin, it is a subspace of dimension 1. Mark all the ones that are subspaces.
If a line doesn’t go through the origin, it is a translation of a line that does. For example, $y = x+2$ is a translation of $y=x$. Translation means sliding it around without changing its slope, or, equivalently, adding a constant to one side of the equation. For each subspace you marked, find out which lines are the translations of that subspace. The set of translations of a line are called its cosets.
Show that a line and a non-trivial translation of that line don’t intersect.
Suppose I send a message, which is an element of $\mathbb{F}_3^2$, i.e. a vector with two entries modulo $3$. Suppose the message is subject to a possible error (at most one) while in transit. Errors can happen in the following way: one of the two entries can accidentally be increased or decreased by one. If I sent $(1,2)$, what are the possible things that could be received? (E.g., $(1,1)$ is possible… there are five possibilities total, including no error.)
Now, continuing the situation above, which system (“code”) should I use to send a bit? I give two possibilities below. Which is the best system and why?
zero = (0,0), one = (0,1)
zero = (0,0), one = (1,1)
Can you come up with a better system to send a bit? What would better mean? (This is open ended.)