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

Abstract

We propose $\mathcal {Q}(q,d)$ , a new family of $\mathbb {F}_{q}$ -ary non-CSS qudit stabilizer codes parameterised by a prime power $q$ and dimension $d\ge 2$ , constructed from Singer difference sets and non-degenerate quadrics in the projective geometry $\mathop {\mathrm {PG}}\nolimits (d,q)$ . The parity-check matrix $H {\,}\text {=}{\,}(A\mid M_{Q}A)$ couples the Singer-circulant point-hyperplane incidence matrix $A$ with the formal trace-of-square quadric circulant $M_{Q}$ , and a Cancellation Lemma exploits the design-theoretic identity $(k-\lambda){\,}\text {=}{\,}q^{d-1}\equiv 0\pmod p$ to make $H$ symplectic-isotropic over $\mathbb {F}_{q}$ for every $(q,d)$ . The novelty of the construction is concentrated in five original structural theorems: (i) an unconditional cross-correlation modular-constancy theorem $\nu \equiv 1\pmod q$ at $q$ odd, $d\ge 3$ with a sharp two-value pattern $V_{2}-V_{1}{\,}\text {=}{\,}q^{(d-1)/2}$ , proving a 2-torsion distance ceiling; (ii) a Chinese-Remainder-Theorem (CRT) zero-count theorem $\#\{f:\tau _{H,f}{\,}\text {=}{\,}0\}{\,}\text {=}{\,}(|\mathcal {F}|-1)/2$ ; (iii) an explicit Clifford bridge $U_{M}{\,}\text {=}{\,}H^{\otimes n}{\,}\mathrm {CZ}^{(M)}{\,}H^{\otimes n}$ that identifies $\mathcal {Q}(q,2)$ as Clifford-equivalent to a graph-state encoding of the trivial Singer CSS code, with the classical 2021 low-density-spreading (LDS) sign-flip pattern realised by the symmetric circulant $M$ ; (iv) a paired column-rigidity equality $d_{\min }{\,}\text {=}{\,}R_{P}+1$ exactly; and (v) a rigorous lower bound $d_{\min }(\mathcal {Q}(7,2))\ge 8$ from an exhaustive Singer-Frobenius search. Via a Hamada-Smith specialisation we obtain the closed-form logical dimension $k(p,d){\,}\text {=}{\,}\theta _{d}(p)-\binom {p+d-1}{d}-1$ , with the flagship $d{\,}\text {=}{\,}2$ family $\mathcal {Q}(p,2){\,}\text {=}{\,}[[p^{2}+p+1,{\,}p(p+1)/2,{\,}d_{\min }]]_{p}$ attaining asymptotic rate $\tfrac {1}{2}$ . For general odd primes $p\ge 3$ the flagship distance is conjectured to satisfy $d_{\min }(\mathcal {Q}(p,2)){\,}\text {=}{\,}p+1$ ; this conjecture is analytically proven exact at $p{\,}\text {=}{\,}5$ and exhaustively lower-bounded to $d_{\min }\ge 8$ at $p{\,}\text {=}{\,}7$ , and remains open for general $p$ . Concrete instances include $\mathcal {Q}(3,2){\,}\text {=}{\,}[[{13,6,3}]]_{3}$ , $\mathcal {Q}(5,2){\,}\text {=}{\,}[[{31,15,6}]]_{5}$ (double-error correcting), $\mathcal {Q}(7,2){\,}\text {=}{\,}[[57,28,d{\ge }8]]_{7}$ , and $\mathcal {Q}(3,4){\,}\text {=}{\,}[[{121,105,3}]]_{3}$ . Monte-Carlo simulations over $1.5{\times }10^{6}$ trials show that $\mathcal {Q}(5,2)$ achieves a $\sim 70\times $ per-logical-qudit improvement over Steane $[[{7,1,3}]]_{2}$ at depolarising rate $p_{\mathrm {ch}}{\,}\text {=}{\,}10^{-3}$ with zero silent logical errors (the “erase-don’t-mistake” property). We use a textbook bounded-distance decoder throughout, leaving Berlekamp-Massey-style cyclic decoding exploiting the Singer-circulant structure as the natural sequel. A fifteen-row axis-by-axis comparison against the closest prior constructions establishes that the Singer-difference-set + quadric-Gram-matrix combination, together with the closed-form $k$ , the cross-correlation and CRT zero-count theorems, the explicit Clifford bridge to the 2021 LDS construction, and the proven 2-torsion ceiling, is new.