Golay codes and quantum contextuality

It is shown that the codewords of the binary and ternary Golay codes can be converted into rays in ${\mathbb{RP}}^{23}$ and ${\mathbb{RP}}^{11}$ that provide proofs of the Kochen-Specker theorem in real state spaces of dimensions 24 and 12, respectively. Some implications of these results are discussed.

Figure 1

CECC = Classical error correcting codes, QECC = Quantum error correcting codes, and Quantum paradoxes = Bell and KS theorems. The connection indicated by arrow 1 was made in many ways, notably by Steane [15] in his seven-qubit code and later in the CSS codes [17]. The connection 3 was made by Shor [7], who showed how the GHZ-Mermin proof could be made to yield quantum codewords, while several examples of the reverse connection 2 were pointed out by DiVincenzo and Peres [8]. The connection 4 was made in this paper, with the dashed arrow being used because we are aware of only two instances of it (based on the binary and ternary Golay codes).