Binary antipodal code

Cousins

• Binary code— Binary antipodal codes are spherical codes obtained from binary codes via the antipodal mapping.
• Slepian group-orbit code— Any length-\(n\) binary linear code can be used to define a diagonal subgroup of \(n\)-dimensional rotation matrices with \(\pm 1\) on the diagonals via the antipodal mapping \(0\to+1\) and \(1\to-1\). The orbit of this subgroup yields the corresponding Slepian group-orbit code; see [1; Thm. 8.5.2].
• Biorthogonal spherical code— Each first-order RM\((1,m)\) code maps to a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping [2][3; Sec. 6.4][1; pg. 19]. In other words, first-order RM (biorthogonal spherical) codes form orthoplexes in Hamming (Euclidean) space.
• Hypercube code— Binary antipodal codes are subcodes of a hypercube code since the hypercube code corresponds to the Hamming \(n\)-cube (a.k.a. Boolean hypercube) embedded into the unit \(n\)-sphere.
• Simplex spherical code— Binary simplex codes map to \((2^m,2^m+1)\) simplex spherical codes under the antipodal mapping [3; Sec. 6.5.2][1; pg. 18]. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.