The q = 2 Singer-quadric qubit codes have minimum distance 2 for every d >= 2

Abstract

Kulhandjian and Hanzo (arXiv:2610.02392) build qubit stabilizer codes Q(2,d) from a Singer difference set and the quadric Tr(z^3) = 0 in PG(d,2), prove numerically for d <= 6 that the cross-correlation u = M_Q tau_H is the all-ones vector, and conjecture (Conjecture 24) that this and the resulting minimum distance 2 hold for every d >= 2. For every d >= 2 and every primitive element of GF(2^(d+1)), u is all-ones, every Z_i Z_j with i != j lies in the centralizer and not in the stabilizer row space, and no element of the centralizer outside the stabilizers has weight below 2.

Definition 1.1 (The field).

$$\forall d:\mathbb{N},K\left(d\right)=\mathrm{GaloisField}\left(2,d+1\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.K (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

K(d) is the Galois field GF(2^(d+1)), the field K of the paper at q = 2.

Definition 1.2 (The code length).

$$\forall d:\mathbb{N},n\left(d\right)=2^{d+1}-1$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.n (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The number of qubits n = (q^(d+1) - 1)/(q - 1) of the paper at q = 2, which is 2^(d+1) - 1, the order of the multiplicative group of K(d).

Definition 1.3 (The absolute trace).

$$\forall d:\mathbb{N},\forall x:K\left(d\right),\mathrm{tr}\left(x\right)=\mathrm{trace}\left(\mathrm{ZMod}\left(2\right),K\left(d\right),x\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.tr (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The trace of K(d) over its prime field ZMod 2, the trace Tr from K to F of the paper.

Definition 1.4 (The hyperplane indicator).

$$\forall d:\mathbb{N},\forall \alpha :K\left(d\right),\forall i:\mathrm{Fin}\left(n\left(d\right)\right),\mathrm{tauH}\left(\alpha \right)\left(i\right)=\mathrm{ite}\left(\mathrm{tr}\left({\alpha }^i\right)=0,1,0\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.tauH (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Appendix A, Step 2: the i-th entry is 1 when the trace of alpha^i vanishes and 0 otherwise, for i in Fin(n(d)). Here ite(c, a, b) is a if c holds and b otherwise.

Definition 1.5 (The quadric indicator).

$$\forall d:\mathbb{N},\forall \alpha :K\left(d\right),\forall i:\mathrm{Fin}\left(n\left(d\right)\right),\mathrm{tauQ}\left(\alpha \right)\left(i\right)=\mathrm{ite}\left(\mathrm{tr}\left({\alpha }^{3\cdot i}\right)=0,1,0\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.tauQ (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Appendix A, Step 3 with xi = q + 1 = 3: the i-th entry is 1 when the trace of alpha^(3i) vanishes and 0 otherwise.

Definition 1.6 (The Z block).

$$\forall d:\mathbb{N},\forall \alpha :K\left(d\right),\mathrm{Hz}\left(\alpha \right)=\mathrm{circulant}\left(\mathrm{tauH}\left(\alpha \right)\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.Hz (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

H_z = A is the Singer incidence matrix circ(tau_H). Mathlib’s circulant(v) has entries v(i - j), with subtraction in Fin(n(d)), the paper’s (M){i,j} = v{(i-j) mod n}.

Definition 1.7 (The X block).

$$\forall d:\mathbb{N},\forall \alpha :K\left(d\right),\mathrm{Hx}\left(\alpha \right)=\mathrm{circulant}\left(\mathrm{tauQ}\left(\alpha \right)\right)\cdot \mathrm{circulant}\left(\mathrm{tauH}\left(\alpha \right)\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.Hx (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

H_x = M_Q A, the product of the quadric circulant M_Q = circ(tau_Q) and A, with matrix multiplication over ZMod 2.

Definition 1.8 (The centralizer).

$$\forall d:\mathbb{N},\forall \alpha :K\left(d\right),\forall a:\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right),\forall b:\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right),(a,b)\in \mathrm{centralizer}\left(\alpha \right)\iff (\mathrm{vecMul}\left(a,\mathrm{transpose}\left(\mathrm{Hx}\left(\alpha \right)\right)\right)=\mathrm{vecMul}\left(b,\mathrm{transpose}\left(\mathrm{Hz}\left(\alpha \right)\right)\right))$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.centralizer (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The paper’s centralizer: pairs (a, b) of vectors over ZMod 2 indexed by Fin(n(d)) with a H_x^T = b H_z^T, where a is the Z part and b the X part and vecMul(c, M) is the row vector c times M.

Definition 1.9 (The stabilizer row space).

$$\forall d:\mathbb{N},\forall \alpha :K\left(d\right),\forall a:\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right),\forall b:\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right),(a,b)\in \mathrm{stabilizers}\left(\alpha \right)\iff (\exists c:\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right),(a=\mathrm{vecMul}\left(c,\mathrm{Hz}\left(\alpha \right)\right))\wedge (b=\mathrm{vecMul}\left(c,\mathrm{Hx}\left(\alpha \right)\right)))$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.stabilizers (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The row space of H = (H_z | H_x): the pairs (c H_z, c H_x) for all row vectors c over ZMod 2.

Definition 1.10 (The symplectic weight).

$$\forall d:\mathbb{N},\forall a:\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right),\forall b:\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right),\mathrm{wt}\left((a,b)\right)=\mathrm{card}\left(\mathrm{filter}\left(i\mapsto (a\left(i\right)\ne 0)\vee (b\left(i\right)\ne 0),\mathrm{univ}\left(\mathrm{Fin}\left(n\left(d\right)\right)\right)\right)\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.wt (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The number of positions i at which (a_i, b_i) is not (0, 0), the weight of a Pauli operator. For the weight-two witness it equals the Hamming weight of (a, b), and every lower bound for it is a lower bound for that Hamming weight.

Definition 1.11 (Conjecture 24).

$$claim\iff (\forall d:\mathbb{N},(2\le d)\Rightarrow (\forall \alpha :K\left(d\right),(\mathrm{orderOf}\left(\alpha \right)=n\left(d\right))\Rightarrow ((\mathrm{mulVec}\left(\mathrm{circulant}\left(\mathrm{tauQ}\left(\alpha \right)\right),\mathrm{tauH}\left(\alpha \right)\right)=1)\wedge ((\forall i:\mathrm{Fin}\left(n\left(d\right)\right),\forall j:\mathrm{Fin}\left(n\left(d\right)\right),(i\ne j)\Rightarrow (((\mathrm{single}\left(i,1\right)+\mathrm{single}\left(j,1\right),0)\in \mathrm{centralizer}\left(\alpha \right))\wedge (\neg ((\mathrm{single}\left(i,1\right)+\mathrm{single}\left(j,1\right),0)\in \mathrm{stabilizers}\left(\alpha \right)))))\wedge (\forall v:(\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right))\times (\mathrm{Fin}\left(n\left(d\right)\right)\to \mathrm{ZMod}\left(2\right)),(v\in \mathrm{centralizer}\left(\alpha \right))\Rightarrow ((\neg (v\in \mathrm{stabilizers}\left(\alpha \right)))\Rightarrow (2\le \mathrm{wt}\left(v\right))))))))$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.claim (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Proposition 23 for every d >= 2 and every primitive element alpha (an element of order n(d)): u = M_Q tau_H is the all-ones vector; for every pair of distinct positions i, j the operator Z_i Z_j, written (e_i + e_j, 0), lies in the centralizer and not in the stabilizer row space; and every element of the centralizer outside the stabilizers has weight at least 2. Together these give d_min = 2.

Theorem 1.12 (The q = 2 distance ceiling holds for every d >= 2).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.result (✓ std3). ∎

Resolves. Problems/kulhandjian-hanzo-2026-q2-distance-ceiling (proved) by D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.result.

Source. Repository-derived.

Acknowledgement. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Put m = d + 1 >= 3 and n = 2^m - 1 >= 7. Power sums over the units of K(d) vanish except when n divides the exponent, and no exponent 2^r, 3 2^s or 3 2^s - 2^r with r, s < m is divisible by n: the last would give 2^t = 3 modulo n with t < m. Writing the trace as the sum of the Frobenius powers x^(2^i), the entry u_k is the sum over x in the units of (1 + Tr(x))(1 + Tr(alpha^(3k) x^(-3))); every non-constant term is a multiple of a vanishing power sum and the constant term is n = 1 in ZMod 2, so u is all-ones and H_x is the all-ones matrix. For b != 0, x -> Tr(b x) is a nonzero linear functional, so each of its fibers has 2^(m-1) elements. Every vector in the row space of A has entries c + Tr(b alpha^(-j)), so its weight is 0, n, 2^(m-1) or 2^(m-1) - 1, never 2; hence Z_i Z_j is not a stabilizer, while H_x = J puts it in the centralizer. A centralizer element of weight 1 would make the all-ones vector, a column of A, or its complement zero, but every column of A has weight 2^(m-1) - 1, strictly between 0 and n.

References

• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.Hx
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.Hz
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.K
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.centralizer
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.claim
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.n
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.result
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.stabilizers
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.tauH
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.tauQ
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.tr
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.wt