\([[16,4,4]]\) biplane code[1]
Description
Self-dual pure CSS code whose \(X\)- and \(Z\)-type stabilizer generators are both supported on the \([16,6,6]\) binary code spanned by the blocks of the biplane of order four, i.e., the symmetric \(2\)-\((16,6,2)\) design whose automorphism group is the affine symplectic group \(\mathbb{Z}_{2}^{4}\rtimes Sp(4,\mathbb{Z}_2)\) of order \(11520\). Equivalently, the underlying classical code is the first-order RM code \(\text{RM}(1,4)\) enlarged by the indicator vector of an elliptic quadric in \(\mathbb{F}_2^4\), so that the code is obtained from the tesseract color code by adding one \(X\)-type and one \(Z\)-type stabilizer generator supported on that quadric. The code admits weight-six stabilizer generators of both types and realizes its full logical Clifford group using depth-one two-local physical circuits.
Labeling the qubits by the points of \(\mathbb{F}_2^{4}\) in binary order, the first five generators of each type are the affine functions spanning \(\text{RM}(1,4)\) and the sixth is the zero set \(\{x: Q(x)=0\}\) of the elliptic form \(Q=x_1x_2+x_3+x_3x_4+x_4\), giving the stabilizer tableau \begin{align} \begin{smallmatrix} X & X & X & X & X & X & X & X & X & X & X & X & X & X & X & X \\ I & X & I & X & I & X & I & X & I & X & I & X & I & X & I & X \\ I & I & X & X & I & I & X & X & I & I & X & X & I & I & X & X \\ I & I & I & I & X & X & X & X & I & I & I & I & X & X & X & X \\ I & I & I & I & I & I & I & I & X & X & X & X & X & X & X & X \\ X & X & X & I & I & I & I & X & I & I & I & X & I & I & I & X \\ Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z \\ I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z \\ I & I & Z & Z & I & I & Z & Z & I & I & Z & Z & I & I & Z & Z \\ I & I & I & I & Z & Z & Z & Z & I & I & I & I & Z & Z & Z & Z \\ I & I & I & I & I & I & I & I & Z & Z & Z & Z & Z & Z & Z & Z \\ Z & Z & Z & I & I & I & I & Z & I & I & I & Z & I & I & I & Z \end{smallmatrix}~. \tag*{(1)}\end{align} Rows 1–5 and 7–11 are the tesseract color-code generators, while rows 6 and 12 are the added quadric generators. The sixteen weight-six codewords of the underlying classical code are the blocks of the biplane; each block contains six qubits and each qubit lies in six blocks, and any six linearly independent blocks form an alternative generating set consisting entirely of weight-six checks.
The permutation automorphism group of the underlying classical code is the group of affine symplectic maps \(x\mapsto Mx+a\) of \(\mathbb{F}_2^4\), where \(M\) preserves the polar form of \(Q\). Such maps preserve \(\text{RM}(1,4)\) and send \(Q\) into its own coset, and there are \(2^4\cdot|Sp(4,\mathbb{Z}_2)|=11520\) of them.
Transversal and Permutation-Based Gates
CNOT gate between two blocks because the code is CSS, and transversal Hadamard because the code is self-dual.Transversal \(S\) and \(\sqrt{X}\) are valid logical gates up to Pauli corrections because every codeword of the underlying classical code has even weight.Fold-transversal gates built from involutions of the order-\(11520\) affine symplectic automorphism group, such as the \(15\) fixed-point-free translations \(x\mapsto x+a\) of \(\mathbb{F}_2^4\), each of which contributes a layer of eight \(CZ\) gates [1].The transversal and fold-transversal gates above, all of which are depth-one two-local physical Clifford circuits, generate the full logical Clifford group \(Sp(8,\mathbb{Z}_2)\) on the four logical qubits [1].Cousins
- \([[16,6,4]]\) Tesseract color code— The \([[16,4,4]]\) biplane code is obtained from the tesseract color code by adding one \(X\)-type and one \(Z\)-type stabilizer generator supported on an elliptic quadric of \(\mathbb{F}_2^4\), reducing the number of logical qubits from six to four. The promoted generators are a minimum-weight representative of one of the \(28\) logical classes of the tesseract code, which is why the resulting code remains pure.
- Quantum Reed-Muller (RM) code— The underlying classical code of the \([[16,4,4]]\) biplane code is the first-order RM code \(\text{RM}(1,4)\) enlarged by the indicator vector of an elliptic quadric in \(\mathbb{F}_2^4\).
- \((16,256,6)\) Nordstrom-Robinson (NR) code— The underlying \([16,6,6]\) classical code of the \([[16,4,4]]\) biplane code is, up to a coordinate permutation, a linear subcode of the NR code, consisting of \(\text{RM}(1,4)\) together with one of the seven nontrivial cosets of \(\text{RM}(1,4)\) making up the NR code.
- Combinatorial design— The sixteen minimum-weight codewords of the underlying \([16,6,6]\) classical code of the \([[16,4,4]]\) biplane code are the blocks of the symmetric \(2\)-\((16,6,2)\) design, i.e., the biplane of order four whose automorphism group is \(\mathbb{Z}_{2}^{4}\rtimes Sp(4,\mathbb{Z}_2)\).
- \([[16,4,4]]\) twisted color code— The \([[16,4,4]]\) biplane code and the \([[16,4,4]]\) symplectic-double code are the two self-dual CSS gauge fixings of the \([[16,4,2,4]]\) tesseract subsystem code [2], obtained by promoting the weight-six and the weight-four gauge operators, respectively.
- Quadric tower code— Both the \([[16,4,4]]\) biplane code and the quadric tower codes are self-dual CSS codes built from quadratic forms over \(\mathbb{F}_2\), the former using a quadric as an extra stabilizer generator and the latter using a quadric as the qubit set.
Primary Hierarchy
References
- [1]
- V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
- [2]
- B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
Page edit log
- Victor V. Albert (2026-08-08) — most recent
Cite as:
“\([[16,4,4]]\) biplane code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/biplane_16_4_4, arXiv:2606.11484