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 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…