Quadric tower code[1]

Description

Family of self-dual CSS codes, one for each \(j\geq 3\), whose qubits are the points of an elliptic quadric in \(\mathbb{F}_2^{2j}\) and whose \(X\)- and \(Z\)-type stabilizer generators are both supported on the order-\((j-2)\) RM code punctured to that quadric. The qubit permutation automorphism group contains the symplectic group \(Sp(2j,\mathbb{Z}_2)\) acting in its minimal-degree faithful permutation representation.

Fix an elliptic quadratic form \(Q_0\) on \(V=\mathbb{F}_2^{2j}\), i.e., one of Arf invariant one, and take its zero set \(Z(Q_0)=Q_0^{-1}(0)\) as the qubits. Both classical codes are the restrictions to \(Z(Q_0)\) of all Boolean functions of algebraic degree at most \(j-2\), \begin{align} C_X=C_Z=\text{RM}(j-2,2j)\big|_{Z(Q_0)}~, \tag*{(1)}\end{align} yielding the family \begin{align} QT_j=[[2^{2j-1}-2^{j-1},\,k_j,\,2^{j-1}]]~,\qquad k_j=14,48,166,584,2092,\ldots \tag*{(2)}\end{align} for \(j=3,4,5,6,7,\ldots\), with \(k_j=V_{j-1}-V_{j-2}\) in terms of the alternating binomial sums \(V_r=S_r-S_{r-2}+S_{r-4}-\cdots\) and \(S_a=\sum_{i\leq a}\binom{2j}{i}\). The first two members are \(QT_3=[[28,14,4]]\) and \(QT_4=[[120,48,8]]\).

Writing \(B\) for the polar form of \(Q_0\), the map \(a\mapsto Q_0+B(a,\cdot)\) identifies the qubits with one Arf class of quadratic forms on \(\mathbb{F}_2^{2j}\). The symplectic group permutes that class, acting affinely on the quadric coordinates, and the class is the smallest set on which \(Sp(2j,\mathbb{Z}_2)\) can act faithfully [2,3]. For \(j=3\), the classical code is the \([28,7,12]\) code whose automorphism group is \(Sp(6,\mathbb{Z}_2)\) [4].