Hadamard Codes and Sphere Packing

Levent Alpöge and colleagues used Claude AI to discover a new Hadamard matrix, prompting a series of posts by John D. Cook on Hadamard matrices and their applications, including NASA's use of a Hadamard matrix of order 32 in the Mariner 9 spacecraft in 1971. Cook's latest post describes how a (8, 4, 4) Hadamard code yields the E8 sphere packing, the densest packing in ℝ8, via Construction A from Conway and Sloane.

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 https://www.johndcook.com/blog/2026/08/13/constructing-hadamard-matrices/ 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 https://www.johndcook.com/blog/2026/08/13/mariner-hadamard/ 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 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 E 8, the densest packing in ℝ8. We start with the Hadamard matrix and obtain the matrix 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.