8 - Other aspects of coding theory

Summary

We end this introduction to coding and information theory by giving two examples of how coding theory relates to quite unexpected other fields. Firstly we give a very brief introduction to the relation between Hamming codes and projective geometry. Secondly we show a very interesting application of coding to game theory.

Hamming code and projective geometry

Though not entirely correct, the concept of projective geometry was first developed by Gerard Desargues in the sixteenth century for art paintings and for architectural drawings. The actual development of this theory dated way back to the third century to Pappus of Alexandria. They were all puzzled by the axioms of Euclidean geometry given by Euclid in 300 BC who stated the following.

1. (1) Given any distinct two points in space, there is a unique line connecting these two points.

2. (2) Given any two nonparallel lines in space, they intersect at a unique point.

3. (3) Given any two distinct parallel lines in space, they never intersect.

The confusion comes from the third statement, in particular from the concept of parallelism. How can two lines never intersect? Even to the end of universe?

In your daily life, the two sides of a road are parallel to each other, yet you do see them intersect at a distant point. So, this is somewhat confusing and makes people very uncomfortable. Revising the above statements gives rise to the theory of projective geometry.