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Let G be the automorphism group of the four-dimensional cube, a group of order $2^4 \cdot 4! = 384$. The binary codes associated with the 32-dimensional permutation representation of G on the edges of the cube are investigated. There are about 400 such codes, one of which is a $[32, 17, 8]$ code, having twice as many codewords as the $[32, 16, 8]$ extended quadratic residue code.

Codes From Symmetry Groups, and a $[32, 17, 8]$ Code