A Venn Diagram of the Hamming Code

Introduction.

The easiest way to describe the (extended) Hamming code, perhaps, is to say that it is a particular 4-dimensional subspace of F28. Since the field F2={0,1} of scalars has only 2 elements, every vector corresponds naturally to a subset of the numbers U={1,2,3,4,5,6,7,8}. Thus, the value 1 appears in the ith coordinate of a vector v if i appears in the set corresponding to v and 0 appears in the ith coordinate if i is not in the corresponding set. Thus, subspaces correspond to subsets of the power set P(U), and it does just as well to report elements of the Hamming code as such. Vector addition translates to the symmetric-difference operation.

As it turns out, there is a Venn diagram depicting the Hamming code with 7-fold rotational symmetry.

There are many distinct imbeddings of the Hamming code in F28. Here we prefer to use that spanned by the 4 codewords

{1,2,3,8}, {1,4,5,8}, {1,6,7,8}, and {3,5,6,8}.

One may conveniently list the 16 codewords as
{},

1238, 1458, 1678, 2468, 2578, 3478, 3568,

4567, 2367, 2345, 1357, 1346, 1256, 1247,

U.

Notice all the usual properties of the Hamming code. For example, any two weight-4 codewords have either 0, 2, or 4 coordinates in common. The weight ennumerator polynomial is
x8+14x4+1,

and so on.

The Venn Diagram.

Figure 1. A Venn Diagram of the Hamming Code.

In the diagram given, all 16 codewords are present, each represented by some connected region in the plane, and every non-bounding point in the plane is represented by a single codeword. The simply-connected region demarked by the heavy lines represents the union of all the codewords containing the number 1. Next, notice that rotating the diagram by 1/7th of a circle is equivalent to applying the permutation (1245736) to the codewords. Thus, one should be able to find six more congruent regions, subsequently representing those regions containing 2, 4, 5, and so on. Finally, there is one jagged disc which is invariant by this rotation, the union of all those regions containing the number 8. Notice that the universal set U=12345678 lies in the middle.

Further Speculations.

Naturally, after seeing this picture and some variants of it, one might wish to consider other linear binary codes. The Golay code clearly stands out as a goal. There is some numerological evidence that there should be a similar thing for the Golay code. Recall that the Golay code has 4096 codewords and that its automorphism group is the the Mathieu group M24. The Mathieu group has a subgroup of order 23, which is necessarily cyclic and it also happens that 23 is a factor of 4096-2=4094. One subtracts 2 from 4096 to account for the empty set and the universal set, just as was the case for the Hamming code. Next, notice that the Golay code is a particular subspace of F224. Recall that with the Venn diagram of the Hamming code, there was a central circle for the number 8. Now notice that 24-1=23, and again we see the number 23. My guess is that one should seek a Venn diagram for the Golay code which has 23-fold rotational symmetry. In this proposed diagram, there should be 23 regions, all rotationally congruent to one another, and one central region which has 23-fold rotational symmetry, thus accounting for all 24 dimensions of the ambient vector space.

Figure 2. A Variant on the Diagram.

References.

Peter Hamburger. Doodles and doilies, non-simple symmetric Venn diagrams. Discrete Mathematics. 257 (2002) 423-439.

Frank Ruskey and Mark Weston. A Survey of Venn Diagrams