Fano Plane

Fano Plane

Three-dimensional codes and projective planes

Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016

This description is true for any three-dimensional code. In our example the 7 coordinates are described by the 7 points of the Fano plane. As each line has 3 points, there are 4 points outside each line. We have seen that our code is a [7, 3, 4]2 code, more precisely each nonzero word of our code has weight 4.

A note on balanced incomplete block designs and projective geometry

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2021

Ömür Deveci, Anthony G. Shannon

The number of lines incident at a given point is r = n + 1. n = 2 gives a projective plane of order 2, a Fano plane, with v = 7 points and 7 lines (from above). In the Fano plane, each line has n + 1 = 3 points and each point belongs to n + 1 = 3 lines. Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs. There are also random complete block designs and Hadamard designs.

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Three-dimensional codes and projective planes

A note on balanced incomplete block designs and projective geometry