Generalized quantum divisible code[1] 

A level-\(\nu\) generalized quantum divisible code is a CSS code whose \(X\)-type stabilizers, in the symplectic representation, have zero norm and form a \((\nu,t)\)-null matrix (defined below) with respect to some odd-integer vector \(t\) [1; Def. V.1]. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy. Such codes can also be level-lifted [1; Theorem V.6], \(\nu\to\nu+1\), which recursively yields towers of generalized divisible codes from a particular ground code.

Given an odd-integer coefficient length-\(n\) vector \(t\), two vectors \(v,w\) are \((\nu,t)\)-orthogonal if \begin{align} \sum_i v_i t_i w_i \equiv 0 \mod 2^{\nu-1}~. \tag*{(1)}\end{align} A matrix whose rows make up such vectors is called \((\nu,t)\)-orthogonal.