Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q)PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance

I. Introduction

• Q(q, d) combines a Singer-circulant incidence matrix with a quadric matrix to construct non-CSS qudit stabilizer codes over PG(d, q).The paper frames this as an algebraic bridge between finite-geometry quantum coding and classical LDS-NOMA construction.Source: §I · p. 2, A. Application domain and system… · p. 2

A. Application domain and system architecture

• Q(q, d) is proposed for quantum communication, repeater stations, and quantum memory, with stabilizer syndromes obtained through non-destructive Pauli measurements.The described physical platforms include photonic encodings and spin-j atoms.Source: A. Application domain and system… · p. 2, Figure 1 · p. 3

B. Why qudit codes? Advantages, trade-offs, and applications

• Qudit carriers hold log2 q bits, but resolving q levels raises detector-resolution and readout-cross-talk demands compared with binary readout.The text also notes that qudit Clifford-circuit synthesis is more involved, although recent compiler advances have reduced the gap.Source: B. Why qudit codes? Advantages,… · pp. 2–3

C. Prior work

• Prior constructions use Singer difference sets or finite-geometry incidences, but the paper distinguishes Q(q, d) by pairing Singer circulants with a quadric Gram matrix.Nguyen and Kim use two shifts of one difference-set indicator, while Gandhi uses a generic symmetric M; neither uses this pairing.Source: C. Prior work · p. 4, (a) Code-construction novelty · p. 5

D. Novelty assessment

• The stated novelty is a unified construction and structural theory; the comparison reports Q(q, d) checks all fifteen contribution rows while Gandhi checks four.The claimed structural results include modular cross-correlation, CRT zero counts, a Clifford bridge, paired-rigidity distance characterization, and a Q(7, 2) lower bound.Source: D. Novelty assessment · p. 4, (b) Structural-theorem novelty · pp. 5–6

E. Contributions

• The Cancellation Lemma makes H = (A | M_Q A) symplectic-isotropic because the difference-set order q^(d−1) vanishes modulo p.The contribution summary states this applies for every (q, d).Source: (b) Structural-theorem novelty · p. 5, E. Contributions · p. 6
• For odd q and d ≥ 3, the paper gives ν[k] ≡ 1 (mod q) and a two-value cross-correlation pattern, yielding a distance ceiling in the stated odd-d case.It also reports the CRT zero-count identity and paired-rigidity equality d_min = R_P + 1.Source: (b) Structural-theorem novelty · p. 5, E. Contributions · p. 6
• The flagship Q(p, 2) family has asymptotic rate 1/2, while d_min = p + 1 remains a conjecture for general odd primes.The distance is proven at p = 5 and lower-bounded by 8 at p = 7; the paper also reports a Q(5, 2) erase-don’t-mistake property and Monte-Carlo evaluation with a bounded-distance decoder.Source: D. Novelty assessment · p. 4, E. Contributions · p. 6

F. Notation

• F_q denotes the finite field of q elements, K an extension field, and Tr_K/F the trace map; the qudit Pauli group is denoted G_n.The notation section also identifies the Singer incidence matrix A and formal quadric circulant M_Q.Source: F. Notation · pp. 6–7, A. Finite fields and the trace function · p. 7

B. Projective geometry PG(d, q)

• PG(d, q) is the projective geometry of one-dimensional subspaces of F_q^(d+1), with θ_d(q) = (q^(d+1) − 1)/(q − 1) points.Its r-flats are the (r + 1)-dimensional subspaces.Source: F. Notation · p. 6, B. Projective geometry PG(d, q) · p. 7
• The point-hyperplane incidence matrix has θ_d−1(q) ones in each row and column, and distinct hyperplanes share θ_d−2(q) points.The text identifies the resulting matrix relation with the difference-set identity and Singer parameters.Source: B. Projective geometry PG(d, q) · p. 7

C. Singer’s theorem

• Singer’s theorem provides a labeling for which the point-hyperplane incidence matrix is circulant, with its first column specified by a trace-zero condition.The cited theorem states this for a primitive element of F_(q^(d+1)).Source: C. Singer’s theorem · p. 7

D. Non-degenerate quadrics

• For even q the quadric indicator uses Tr(z^(q+1)) = 0, whereas for odd q it uses Tr(z^2) = 0.The text also states that a non-degenerate quadric of the specified size exists for even d.Source: D. Non-degenerate quadrics · p. 7

E. Stabilizer formalism and EAQECCs

• A stabilizer code is the joint +1 eigenspace of an Abelian Pauli subgroup, represented by a parity-check matrix H = (H_z | H_x).The code parameters use k = n − rank_Fq(H), with minimum distance defined through centralizer elements outside the stabilizer.Source: F. Notation · p. 6, E. Stabilizer formalism and EAQECCs · p. 7
• The stabilizer condition requires the symplectic inner product of each pair of check rows to vanish.When H fails this condition, the cited entanglement-assisted framework uses pre-shared ebits; the text gives c = 1/2 rank_Fq(S).Source: E. Stabilizer formalism and EAQECCs · p. 7
• The paper works in characteristic p > 2 for its main nontrivial regime, while noting that the cancellation lemma forces S = 0 trivially when q = 2^t.The stated even-rank property for the symplectic deficit is tied to the characteristic assumptions described in the text.Source: E. Stabilizer formalism and EAQECCs · pp. 7–8

A. Difference-set framework

• In PG(d,q), the Singer set E has parameters k=θ_d−1(q) and λ=θ_d−2(q), with each non-identity group difference covered λ times.The points and hyperplanes are indexed by the cyclic group G of order θ_d(q).Source: A. Difference-set framework · p. 8

B. Waterloo decomposition

• The classical Waterloo decomposition exists if and only if d is even.Its signed vector generates a G-invariant balanced weighing matrix through the regular representation of G.Source: B. Waterloo decomposition · p. 8

C. The construction for d even

• For even d, the construction partitions D using whether |gE ∩ Q| takes the larger or smaller of its two values.The construction requires a non-degenerate quadric with |Q|=|E|; the stated intersection counts include q+1 tangent hyperplanes.Source: C. The construction for d even · p. 8

D. Why the construction breaks for d odd

• For odd d, no non-degenerate quadric of cardinality θ_d−1(q) exists, so the cardinality match needed by the construction fails.The trace-of-square set has cardinality strictly below |E|, and the classical character-theoretic converse rules out any Waterloo decomposition.Source: D. Why the construction breaks for d odd · p. 8

IV. The Waterloo Defect for d Odd

• Over F_q, integer-lattice Waterloo defects vanish under reduction modulo p when p divides qd, so symplectic tests cannot witness them.The passage says this limitation applies to any F_q Pauli stabilizer formalism, not only the Gandhi recipe.Source: §IV · p. 8

A. Setup: the candidate stabilizer matrix

• The candidate stabilizer matrix is H=(A | M_QA); at q=2 its symplectic deficit vanishes trivially for every d.The passage attributes this ceiling to the characteristic-two invariant relation, not to parity generally.Source: A. Setup: the candidate stabilizer… · p. 9

B. The Gandhi-recipe deficit vanishes over Fq

• The cancellation lemma gives zero symplectic deficit for H=(A | M_QA) over F_q when M_Q is circulant with equal row sums and d≥2.The coefficient k−λ=q^(d−1) is zero modulo p, so the resulting code has zero ebit consumption.Source: B. The Gandhi-recipe deficit vanishes… · p. 9

C. Numerical sweep of candidate constructions

• The tested V7 recipe gives c=0 at (3,2) and c=6 at (3,3), but also vanishes at (2,3), so it does not isolate parity.V1 over F_q is always CSS in the tested cases.Source: D. What the data say · p. 9, C. Numerical sweep of candidate… · p. 9

E. Attempt at a Z-coefficient lifting (failed)

• At (q,d)=(3,2), the signed split has diag(WW⊤)=6 rather than qd=9, and the formal quadric's correlation takes {0,1,2} instead of two values.The tested sign assignment is not the Arasu–Pott Waterloo split, and the formal trace-of-square quadric does not satisfy that identity verbatim.Source: D. What the data say · p. 9

D. What the data say

• The Arasu–Pott split uses character identities in Z[ζ_n], while the EAQECC framework accepts F_p-coefficient parity checks, not an integer norm-form deficit.A Singer-cycle automorphism search at (3,2) and (5,2) found no map between the formal trace-of-square quadric and the character-theoretic quadric.Source: D. What the data say · pp. 9–10

F. Attempt at a GF(4)/Hermitian lifting (also failed)

• The GF(4) Hermitian deficit is c_H=40 at (3,3) and c_H=12 at (3,2), producing a full-rank deficit in the former and a spurious even-dimension deficit in the latter.The Hermitian condition is stronger than the F2-symplectic condition, so its converse fails.Source: F. Attempt at a GF(4)/Hermitian… · p. 10

G. Conclusion of approach (ii)

• Three lifting strategies provide evidence, but not a formal proof, that the Waterloo decomposition is incompatible with the F_p-Pauli stabilizer formalism because its integer defect vanishes modulo p.A non-stabilizer EAQECC that respects the integer arithmetic is left for future research.Source: G. Conclusion of approach (ii) · p. 10

H. Pivot to a CSS construction (P1)

• The canonical matrix V1 = (A | M_QA) is symplectic-isotropic for every prime power q and d ≥ 2, defining a possibly trivial CSS stabilizer code.The paper adopts this CSS framing for the remainder of its construction discussion.Source: H. Pivot to a CSS construction (P1) · p. 10, §V · p. 11

I. Theorem statement (under pivot P1)

• Theorem 6 gives a valid F_q-ary stabilizer code from H = (A | M_QA), with minimum distance at least q + 1.For (q,d) = (3,3), it gives [[40,29,≥4]]_3.Source: §I · p. 10

V. Stabilizer Code Construction

• The construction couples Singer incidence A with the formal-quadric circulant M_QA in H = (A | M_QA), whose symplectic isotropy guarantees a valid stabilizer matrix.The algorithm builds A and M_Q, forms H, and determines logical dimension from n − rank(H).Source: §I · p. 10, §V · p. 11, A. Algorithm · p. 11

B. The code is not strictly of CSS nature

• At (q,d) = (3,3), H is not a block-diagonal CSS matrix, but its symplectic condition still holds, making it a valid non-CSS stabilizer code.The passage reports H_z H_x^⊤ = J ≠ 0 mod 3.Source: B. The code is not strictly of CSS… · p. 11

C. Recovering the 2021 LDS at d = 2

• At d = 2, the construction shares the LDS code’s Singer and quadric primitives; an explicit Clifford gate relates Q(q,2) to the trivial Singer CSS code.The paper distinguishes this structural relationship from equality of the codespaces.Source: C. Recovering the 2021 LDS at d = 2 · p. 11

A. Existence and dimension

• For prime p, Hamada–Smith rank specialization gives a closed-form logical-qudit count; the d = 2 distance p + 1 remains conjectural beyond the stated cases.The conjecture is exact at p = 5, while exhaustive search gives d_min ≥ 8 at p = 7; the simplified rank formula is established for t = 1, with general-t analysis left for future work.Source: A. Existence and dimension · pp. 11–12

• For odd q and d ≥ 3, the cross-correlation satisfies ν[k] ≡ 1 (mod q) for every k.For odd d, it takes exactly two values whose difference is q^(d−1)/2.Source: B. Minimum distance: a 2-torsion… · pp. 13–14
• The modular-constancy result has a sharp boundary at d = 2, where the quadratic-form zero counts need not be divisible by q.At (3,2), ν takes all three residues {0,1,2} modulo q.Source: B. Minimum distance: a 2-torsion… · p. 13
• For odd q and odd d ≥ 3, the code has d_min ≤ 2 because a non-stabilizer logical of weight 2 arises from the 2-torsion element.The even length n provides the non-trivial 2-torsion element used in the argument.Source: B. Minimum distance: a 2-torsion… · p. 13
• At (q,d) = (3,3), direct verification establishes d_min(Q(3,3)) = 2, with Z_0Z_20 a non-stabilizer logical.The reported exhaustive search also excludes weight-1 centralizer elements.Source: B. Minimum distance: a 2-torsion… · pp. 14–15

• Q(q,2) and the 2021 LDS construction share the Singer incidence and formal-quadric pattern, but their code spaces occupy different ambient spaces.The LDS matrix is a classical spreading matrix, whereas Q(q,2) is a quantum stabilizer code.Source: C. Connection to the 2021 LDS at d = 2 · p. 15
• In the bridge, the trivial Z-only CSS code uses the same Singer incidence A, and conjugation by U_M maps its stabilizer row span to H = (A | M_QA).The circulant condition A M = M_QA supports this reduction.Source: C. Connection to the 2021 LDS at d = 2 · pp. 15–16
• For odd prime q with ord_n(q) = 3, a symmetric circulant M yields the explicit Clifford gate U_M = H^⊗n CZ(M) H^⊗n relating Q(q,2) to the trivial Singer CSS code.The gate is established under the stated order condition.Source: C. Connection to the 2021 LDS at d = 2 · pp. 15–16
• The formal-quadric vector encodes the 2021 LDS sign-flips, while the quantum construction uses it as the X-block coupling and as the basis for the graph-state-style CZ layer.The graph adjacency M is a symmetric circulant constructed by CRT symmetrization.Source: C. Connection to the 2021 LDS at d = 2 · p. 16

D. LDS columns vs the centralizer of Q(q, 2)

• None of the 15 LDS columns lies in the centralizer of Q(3, 2), so none is a logical Z operator.The Z-only centralizer and Z-only sub-stabilizer coincide; all 12 logical Pauli generators are mixed-Pauli.Source: D. LDS columns vs the centralizer of… · p. 16
• Stacking the 15 LDS columns with the rows of A gives rank 13 over F3, so the columns span a 6-dimensional complement to rowspan(A).This complement's dimension matches k(3, 2) = 6, but that is a rank-counting correspondence, not direct centralizer membership.Source: D. LDS columns vs the centralizer of… · pp. 16–17
• A symplectic basis change can map the 6-dimensional complement into the logical-Z sector, but a physically implementable Clifford with linear gate count remains open.A generic symplectic transformation may require O(n^2) gates.Source: D. LDS columns vs the centralizer of… · p. 17

E. Capacity comparison

• The rate of Q(q, d) tends to 1 as d →∞ at fixed q and also as q →∞ at fixed d.The stated large-d limit follows from k(q, d)/n = 1 − rank_Fq(A)/θ_d(q) → 1.Source: E. Capacity comparison · p. 17
• For d = 2, n and k scale as O(p^2), with k = p(p+1)/2, so the rate approaches 1/2 while conjectured d_min ∼ O(p) makes relative distance shrink as O(1/p).Increasing d at fixed p pushes rate toward 1, but distance need not grow linearly with n; Q(3, 4) has parameters [[121, 105, 3]].Source: E. Capacity comparison · p. 17

A. Trace table over F81

• The trace-zero set in F81 has 26 elements, and reducing modulo F3* yields a Singer first column of weight 13 = θ2(3).Figure 2 shows the indicator of the canonical hyperplane in PG(3, 3).Source: A. Trace table over F81 · p. 17, Figure 2 · p. 17

B. Difference-set verification

• The 40 × 40 circulant A is the PG(3, 3) point-hyperplane incidence matrix, with each row and column containing 13 ones.Its Singer parameters are (n, k, λ) = (40, 13, 4), so distinct hyperplanes intersect in four points and k − λ = 9.Source: B. Difference-set verification · p. 18
• The F3-rank of A is 11, yielding 29 logical qudits from 40 physical qudits without entanglement consumption.The 11 independent stabilizer generators correspond to the syndrome bits read during error correction.Source: B. Difference-set verification · p. 18

C. The Waterloo defect at d = 3

• At (q, d) = (3, 3), the Waterloo quadric column has weight 10 rather than the required 13, exhibiting the Waterloo defect.The reported intersections are |Q_o ∩ E| = 8, |Q_o \ E| = 12, and |E \ Q_o| = 18.Source: C. The Waterloo defect at d = 3 · p. 18

D. Final code parameters at (q, d) = (3, 3)

• Q(3, 3) has parameters [[40, 29, 2]]_3, rate 29/40 = 0.725, and detects but does not correct single errors.Its minimum-weight non-stabilizer logical is a 2-torsion Pauli of weight 2.Source: D. Final code parameters at (q, d) =… · p. 18

E. Other instances of Q(q, d) with computed dmin

• The distance ceiling d_min ≤ 2 applies when n is even; by contrast, Q(5, 2) has distance 6 and Q(7, 2) has distance at least 8.Q(3, 4) is a single-error-correcting instance with parameters [[121, 105, 3]]_3; the q = 2 ceiling also limits binary instances to distance 2.Source: E. Other instances of Q(q, d) with… · pp. 18–19

F. Q(5, 2) = [[31, 15, 6]]5: information-set-decoding (ISD)-confirmed distance

• d_min(Q(5, 2)) = 6 exactly, matching p + 1 for p = 5 after exhaustive search found no centralizer member at weights 1 through 5.The weight-6 witness has support {0, 1, 2, 4, 13, 28} and saturates the stated classical bound.Source: F. Q(5, 2) = [[31, 15, 6]]5:… · p. 19

G. The paired-rigidity-distance theorem

• For any stabilizer code, the minimum distance equals the paired column rigidity index plus one: d_min = R_P + 1.The paired index is sharp, unlike the weaker bound from generic column rigidity.Source: G. The paired-rigidity-distance theorem · pp. 19–20

H. Singer-circulant paired rigidity

• d_min(Q(7, 2)) ≥ 8 follows from an exhaustive Singer-Frobenius canonical search through weight 7, including 1,546,218 canonical orbits at weight 7.The search found no rank deficits at those weights; the exact distance remains conjectured to be 8.Source: H. Singer-circulant paired rigidity · pp. 20–21

I. Structural framework via cyclic-code CRT

• Under the condition −1 ∉ ⟨p⟩ in (Z/nZ)*, the CRT zero-count is exactly (|F| − 1)/2, explained by reciprocal pairing and the Hamada-Smith degree sum.The condition holds for p ∈ {3, 5, 7, 11} by direct verification; when it fails, the stated Clifford bridge need not exist, although the stabilizer code still does.Source: §I · pp. 21–22
• An analytical BCH-style proof of the flagship distance bound remains open; current strongest verified statements rely on separate exhaustive or near-exhaustive searches.The paper leaves a finer CRT frequency count for future work.Source: §I · p. 22
• The support-restriction search for Q(5, 2) takes about 90 seconds across weights 1–6, roughly a 200× speedup over the stated naive weight-4 enumeration estimate.The approach restricts the centralizer calculation to position subsets and enumerates the smaller resulting kernels.Source: §I · p. 22

Prior constructions

• Tang, Bai and Feng use flat-incidence dual containment in PG(d, q), rather than Singer circulants or a quadric Gram matrix.Their smallest cited example is [[18, 10, 2]]_2 over PG(2, 3).Source: J. Comparison with prior… · p. 22
• Gandhi’s Singer-difference-set construction uses a generic symmetric matrix with equal row sums, producing binary codes.The cited example is [[273, 191, 11]]_2 from PG(3, 4).Source: J. Comparison with prior… · p. 22

Distinguishing scope

• Q(q, d) instead uses a quadric-derived matrix and gives q-ary qudit codes for q ≥ 3, a parameter regime not covered by those constructions.Source: J. Comparison with prior… · p. 22

A. Axis-by-axis comparison

• The comparison taxonomy covers construction primitives, code class, and structural properties; Q(q, d) ticks all fifteen listed features, versus four for Gandhi.The feature list includes the quadric Gram matrix, non-CSS qudit status, structural theorems, and double-error correction at Q(5, 2).Source: A. Axis-by-axis comparison · pp. 22–23

B. Parameter comparison at small instances

• Q(5, 2) = [[31, 15, 6]]_5 is the only listed moderate-length instance with dmin = 6, achieving double-error correction at n = 31.The cited single-error-correcting comparators have dmin = 3, while Gandhi’s [[273, 191, 11]]_2 requires a much larger block length.Source: B. Parameter comparison at small… · p. 23, Table 8 · p. 24

C. Distinguishing features

• Gandhi’s construction shares the Singer substrate and (A | M A) recipe, but uses a generic matrix and binary qubits rather than a quadric-derived matrix and q-ary qudits.Source: C. Distinguishing features · p. 23
• Tang–Bai–Feng use flat incidences and qubit codes, while Q(p, 2) extends the reported parameter regime to arbitrary prime p with closed-form parameters.The comparison describes prior PG(d, q) work as tending toward small projective planes and binary codes.Source: C. Distinguishing features · p. 23, D. Novelty summary · p. 23

IX. Numerical Evaluation

• The paper evaluates depolarising-channel performance analytically using standard upper bounds on post-decoding logical error rates.Its channel model applies independent depolarising errors to qudits, and the minimum-distance decoder corrects errors of weight at most floor((dmin − 1)/2) with certainty.Source: §IX · p. 23, A. Channel and decoder model · p. 24

B. Asymptotic behaviour

• The reported leading-order comparison includes Q(q, 2), Steane, and Gandhi at p = 10^-3, while per-logical-qudit rates compare codes with different k.The section reports Q(5, 2)’s advantage across three representative channel rates.Source: B. Asymptotic behaviour · p. 24, C. Per-logical-qudit comparison · p. 24, Table 9 · p. 25

C. Per-logical-qudit comparison

• 3.0 × 10^-7 per logical qudit at pch = 10^-3 gives Q(5, 2) a 70× advantage over Steane and 63× over Gandhi.At pch = 10^-2 its advantage over Steane is 9×; Q(3, 2) has a ∼1.6× advantage over Steane/Gandhi.Source: C. Per-logical-qudit comparison · p. 24, Figure 4 · p. 24, Table 10 · p. 25

E. Hashing-bound positioning

• The Q(p, 2) family asymptotically has rate approaching 1/2, but does not asymptotically achieve the quantum hashing bound.The paper describes it as providing explicit structured codes at moderate finite n.Source: E. Hashing-bound positioning · p. 25

F. Monte-Carlo validation

• The Monte-Carlo simulation uses a bounded-distance syndrome-table decoder, sampling i.i.d. depolarising errors and checking whether the residual is a stabilizer.The syndrome table stores errors up to t, and the number of trials adapts to channel rate.Source: F. Monte-Carlo validation · p. 25

G. Discussion of decoder behaviour

• The Monte-Carlo failure probability tracks the analytical upper bound within ∼5% at pch ≥ 3 × 10^-3.The paper attributes the small overshoot at pch = 10^-3 to sampling variance.Source: G. Discussion of decoder behaviour · pp. 25–26, Table 11 · p. 26
• Q(5, 2) had zero silent logical errors across 1.5 × 10^6 trials; every observed decoder failure was declared uncorrectable.For channel errors of weight at most 3, the residual after decoding has weight at most 5, below dmin = 6, so it is a stabilizer.Source: G. Discussion of decoder behaviour · p. 26
• Q(5, 2)’s pseudothreshold is ≈2.6 × 10^-2, about twice Q(3, 2)’s ≈1.3 × 10^-2 despite its larger block length.Source: G. Discussion of decoder behaviour · p. 26

X. Conclusions and Open Problems

• Berlekamp–Massey-style cyclic decoding and belief-propagation decoding remain proposed follow-up directions, not implemented decoders in this work.The paper identifies coupled Z/X sectors through M_Q as a challenge for a joint syndrome-polynomial approach.Source: G. Discussion of decoder behaviour · p. 26, §X · p. 27

Indicator construction

• The construction takes a prime power q, dimension d ≥ 2, and a primitive element α, then builds a trace table over K.The trace values are T[i] = Tr_K/F(α^i).Source: Appendix A · p. 27
• The hyperplane indicator τ_H marks indices whose trace is zero and has Hamming weight θ_(d−1)(q).Source: Appendix A · p. 27
• The quadric indicator τ_Q uses a multiplier ξ that depends on whether q is even or odd, selecting indices with zero trace.Source: Appendix A · p. 27

Circulant assembly

• M_Q is the n × n circulant whose first column is τ_Q and whose other columns are cyclic shifts.Circulant symmetry makes each row’s Hamming weight equal to the weight of τ_Q.Source: Appendix A · p. 28

Worked example

• For (q, d) = (3, 2), the worked example gives n = 13 and a quadric indicator of weight p + 1 = 4.The resulting M_Q matches the projective-plane formal-quadric matrix of [8, Thm. 4.2], modulo re-indexing.Source: Appendix A · p. 28