Hadamard Codes and Sphere Packing

Yesterday Levent Alpöge announced that he and his colleagues had discovered a new Hadamard matrix using Claude AI. That motivated a post I wrote this morning on how to construct Hadamard matrices. I mentioned in that post that these matrices arise in applications.

This evening I gave an example, describing how NASA used a Hadamard matrix of order 32 to transmit photos from the Mariner 9 spacecraft in 1971. This post will give another application: sphere packing.

Conway and Sloane [1] give a correspondence between binary codes and sphere packings that they call Construction A. Given an (n, M, d) binary code C, center a sphere on a point x if and only if x is a congruent (mod 2) to codeword in C.

Here (n, M, d) means an error correcting code that encodes M bits of data as strings of n bits, with a minimum Hamming distance between code words of d, i.e. all codewords differ in at least d bits.

The previous post described how to create a (32, 6, 16) code by stacking a Hadamard matrix H of order 32 on top of −H and turning −1’s into 0’s. The analogous construction for a (8, 4, 4) Hadamard code gives E₈, the densest packing in ℝ⁸.

We start with the Hadamard matrix

$H_8 = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \end{pmatrix}$

and obtain the matrix

$M = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \end{pmatrix}$

whose centers form the sphere packing.

This doesn’t look like the E8 sphere packing as it is usually presented, but it’s isomorphic.

[1] J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. Springer. 1999.

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