Thing I Learned Today: Finite Projective Geometry

by Brian Shourd on January 23, 2013

Tags: math, coding

Today I learned about a math construct called finite projective geometry in the coding theory course that I’m taking. Maybe you are already familiar with Euclid’s Axioms, where Euclid set down an axiomatic list of properties, and then set to prove a ridiculous amount of things based on those five simple properties.

Finite projective geometry is similer, except that there are only three axioms. We start with three sets:

  • \(L\) - the so-called set of lines
  • \(P\) - the set of points
  • \(I \subset P \times L\) - the incidence set

[If you aren’t familiar with the terminology \(P \times L\), it simply means the collection of pairs \((p,l)\) where \(p \in P\) and \(l \in L\). If both \(P\) and \(L\) are finite, then the number of elements in \(P \times L\) is the number of elements in \(P\) times the number of elements in \(L\).]

If a pair \((p,l)\) is in \(I\), we say that \(p\) belongs to \(l\). Be careful - we haven’t defined what points or lines are, outside of the fact that they are elements of named sets. In particular, you cannot be thinking of lines and points in space, as you usually do. Abandon all intuition, ye who enter here.

These sets \(P\), \(L\), and \(I\) form a finite projective geometry if they satisfy the following axioms:

  1. For every two distinct points, there is exactly one line containing both of them.
  2. For every two distinct lines, there is exactly one point contained in both of them.
  3. There exist some four points such that no three belong to the same line.

That’s it. What are some consequences? There are no parallel lines, according to axiom 2, since every pair of lines must intersect. There can’t be fewer than four points, by axiom 3.

Actually, what is the fewest number of points that we can have? Let’s try four. We have four points, \(a\), \(b\), \(c\), and \(d\). First, by axiom 1, there must be lines \(ab\), \(ad\), and \(ac\). Moreover, none of these are the same line, else they would contain three points, violating axiom 3.

Four points, three lines

Four points, three lines

Similarly, we must have lines \(bd\), \(cd\), and \(bc\), each different.

Four points, six lines

Four points, six lines

Are we done? Not quite, we aren’t currently satisfying axiom 2, since the lines \(bd\) and \(ac\) don’t intersect (they look like they do, but notice that there isn’t a single point contained in both of them). We’re stuck - we can’t extend any of our lines without violating axiom 3, and we can’t add any more lines without violating axiom 1. The only thing we could do is add more points.

So there aren’t any examples of a finite projective geometry with four points.

We can add a point \(e\) on the intersection of \(ac\) and \(bd\). But \(ab\) and \(cd\) don’t cross, and neither do \(bc\) and \(ad\). So we’ll add two more points, \(g\) and \(f\), and extend these lines.

Seven points, six lines

Seven points, six lines

Are we done yet? No! There is no line connecting \(f\) and \(g\), or \(f\) and \(e\), or \(e\) and \(g\). So we add a line containing all of them.

Seven points, seven lines

Seven points, seven lines

And it looks like we’re done! Every pair of points is connected by a line, every pair of lines intersect at exactly one point, and there are four points (\(a\), \(b\), \(c\), and \(d\)) of which no three are colinear. We can re-arrange this for a slightly more appealing shape.

Fano plane

Fano plane

This is called the Fano plane, and aside from being pretty cool to look at, it actually has a use in coding theory.

We can rename these points again with numbers in binary from 1 to 7.

Fano plane, binary

Fano plane, binary

Notice that if we perform bitwise addition on any two numbers on a line, we get the third. This is actually the three dimensional vector space over the field of characteristic two (excluding the origin). The basis vectors are \(001\), \(100\), and \(010\) - the corners of the triangle. Of course, we could draw this triangle other ways, and we could also use a different basis.

How does this give us a code? Well, we form what is called the incidence matrix, the matrix \(A\) whose \(i,j\) entry is just one if point \(i\) belongs to line \(j\), and zero otherwise. The columns of this matrix are essentially binary strings of length 7, three of which are 1 and four of which are zero. We’ll call these columns \(v_1, \ldots, v_7\). Each of these has a complement, where we swap all the \(1\)s for \(0\)s and vice versa, which we will call \(\bar{v_1}, \ldots \bar{v_7}\). If we also include the string of all zeros and the string of all ones, we get 16 total strings.

Here’s what’s neat: this particular code is a perfect code. Loosely, that means that it fits the vectors together in such a way that it is easy to correct errors in transmitting the code. Indeed, if we consider the distance between two vectors to be the number of bit flips necessary to get from one to another, then every two vectors of the 16 I just described are at least 3 apart.

What do I mean? Imagine that you have 16 different signals that you want to be able to send. You can encode these digitally as strings of 1s and 0s, but you know that the channel that you send your data over might just corrupt your data (flip some bits). If you encode it with four binary digits, then a single bit flip will result in your signal being decoded as the wrong signal at the other end. But, if you map each signal to one of the 16 vectors given above, and then transmit these (now 10 bits of data), it will take no fewer than three bit flips before a given signal is transformed into a different signal.

Better still - if your channel is reliable enough that you don’t expect a lot of bit flips, you can set it up so that your reciever can correct a single bit flip without needing to retransmit the data.

The fact that this code is perfect means that it occupies some kind of local maximum for packing this error-detecting-correcting capability into a small number of bits, relative to the number of words that you want to send (16) and the number of characters in your alphabet (2). It’s very interesting stuff.