Galois-qudit USt code[1–5]

Description

A Galois-qubit code whose codespace consists of a direct sum of a Galois-qubit stabilizer codespace and one or more of that stabilizer code's error spaces.

Given a subset \(T\) of coset representatives of \(\mathsf{N}(\mathsf{S})/\mathsf{S}\) of a Galois-qudit stabilizer code \(((n,K))\) with codespace \(\mathsf{C}\) and stabilizer group \(\mathsf{S}\), one can construct the Galois-qudit USt with codespace \begin{align} \mathsf{C}_{\text{USt}}=\bigoplus_{t\in T}t\mathsf{C}~. \tag*{(1)}\end{align} The parameters of the USt are \(((n,K|T|,d))\), where \(|T|\) is the number of chosen coset representatives. A Galois-qudit USt is CSS-like when the underlying stabilizer code is CSS, so the coset representatives from the two classical codes underlying the CSS code.

Union stabilizer codes generalize stabilizer codes by modifying the original stabilizer code projection with elements of a subset \(\mathsf{B}\subset\mathsf{S}\) called the Fourier description [4; Thm. 2.7]. When \(\mathsf{B}\) is a subgroup of \(\mathsf{S}\), then the code reduces to an ordinary stabilizer code.

The \(((n, \lceil\frac{q^n}{n(q^2-1)}\rceil,2))_q\) family of Galois-qudit non-stabilizer codes is constructed in Ref. [4].