Galois-qudit non-stabilizer code[1]

The projection onto a stabilizer code is proportional to an equal sum over all elements of the stabilizer group \(\mathsf{S}\). Non-stabilizer codes generalize stabilizer codes by modifying the code projection with elements of a subset \(\mathsf{B}\subset\mathsf{S}\) called the Fourier description (see proof of Thm. 2.7 in Ref. [1]). When \(\mathsf{B}\) is a subgroup of \(\mathsf{S}\), then the code reduces to an ordinary stabilizer code.

The following non-stabilizer codes were constructed in Ref. [1]: \(((33, 155, 3))\), \(((15, 8, 3))\), \(((n, \lceil\frac{q^n}{n(q^2-1)}\rceil,2))_q\) and \(((n, 1+n(q-1),2))_q\), where \(n\) is odd. The last code family is a Galois-qudit extension of the non-additive six-qubit CWS code.