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

Abstract

We propose Q(q,d), a family of q-ary non-CSS qudit stabilizer codes with prime power q and dimension d >= 2, constructed from Singer difference sets and non-degenerate quadrics in PG(d,q). The parity check H = (A | M_Q A) couples the Singer-circulant incidence matrix A with the quadric circulant M_Q, and a Cancellation Lemma uses the identity (k - lambda) = q^{d-1} = 0 mod p to make H symplectic-isotropic over F_q. Five structural theorems carry the novelty: a cross-correlation modular-constancy identity nu = 1 mod q at q odd, d >= 3, with a two-value pattern that proves a 2-torsion distance ceiling; a CRT zero-count identity; an explicit Clifford bridge identifying Q(q,2) as a graph-state encoding of the trivial Singer CSS code, with the 2021 low-density-spreading (LDS) sign-flip pattern realised by a symmetric circulant; a paired column-rigidity equality d_min = R_P + 1; and a lower bound d_min(Q(7,2)) >= 8 from an exhaustive Singer-Frobenius search. A Hamada-Smith specialisation yields the closed-form logical dimension k(p,d) = theta_d(p) - C(p+d-1, d) - 1, and the flagship Q(p,2) = [[p^2 + p + 1, p(p+1)/2, d_min]]_p attains asymptotic rate one-half. We conjecture d_min(Q(p,2)) = p + 1 for odd primes, proven at p = 5 and lower-bounded to >= 8 at p = 7. Instances include Q(3,2) = [[13,6,3]]_3, Q(5,2) = [[31,15,6]]_5, Q(7,2) = [[57,28, d>=8]]_7, and Q(3,4) = [[121,105,3]]_3. Monte-Carlo over 1.5 x 10^6 trials shows Q(5,2) achieves a 70-fold per-logical-qudit improvement over Steane [[7,1,3]]_2 at depolarising rate 10^-3 with zero silent logical errors. A textbook bounded-distance decoder is used; Berlekamp-Massey-style cyclic decoding is left as the natural sequel.