\(k\)-orthogonal code[1–3] 

Qubit stabilizer code whose \(X\)-type logicals and generators form a \(k\)-orthogonal matrix (defined below) in the symplectic representation. In other words, the overlap between any \(k\) \(X\)-type code-preserving Paulis (including the identity) is even. The original definition is for qubit CSS codes [1], but it can be extended to more general qubit stabilizer codes [3; Def. 1]. This entry is formulated for qubits, but an extension exists for modular qudits [1].

A matrix is \(k\)-orthogonal [3; Def. 4] if \begin{align} |x^1|&\equiv 0 \mod 2 \tag*{(1)}\\ |x^1\cdot x^2|&\equiv 0 \mod 2 \tag*{(2)}\\ |x^1\cdot x^2\cdot x^3|&\equiv 0 \mod 2 \tag*{(3)}\\ &\vdots \tag*{(4)}\\ |x^1\cdot x^2\cdot x^3\cdot\ldots\cdot x^k|&\equiv 0 \mod 2 \tag*{(5)}\\ \end{align} for all its rows \(x^j\), where the generalized dot-product notation means a sum of products of the respective coordinates of all vectors.