The existence of optimal quaternary [28; 20; 6] and quantum [[28; 12; 6]] codes

The existence of a quantum [[28; 12; 6]] code was one of the few cases for codes of length n ≤30 that was left open in the seminal paper by Calderbank, Rains, Shor, and Sloane [2]. The main result of this paper is the construction of the first optimal linear quaternary [28; 20; 6] code which contains its Hermitian dual code and yields the first optimal quantum [[28; 12; 6]] code.