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

Abstract

We propose <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(q,d)$ </tex-math></inline-formula>, a new family of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>-ary non-CSS qudit stabilizer codes parameterised by a prime power <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and dimension <inline-formula> <tex-math notation="LaTeX">$d\ge 2$ </tex-math></inline-formula>, constructed from Singer difference sets and non-degenerate quadrics in the projective geometry <inline-formula> <tex-math notation="LaTeX">$\mathop {\mathrm {PG}}\nolimits (d,q)$ </tex-math></inline-formula>. The parity-check matrix <inline-formula> <tex-math notation="LaTeX">$H {\,}\text {=}{\,}(A\mid M_{Q}A)$ </tex-math></inline-formula> couples the Singer-circulant point-hyperplane incidence matrix <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> with the formal trace-of-square quadric circulant <inline-formula> <tex-math notation="LaTeX">$M_{Q}$ </tex-math></inline-formula>, and a Cancellation Lemma exploits the design-theoretic identity <inline-formula> <tex-math notation="LaTeX">$(k-\lambda){\,}\text {=}{\,}q^{d-1}\equiv 0\pmod p$ </tex-math></inline-formula> to make <inline-formula> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula> symplectic-isotropic over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> for every <inline-formula> <tex-math notation="LaTeX">$(q,d)$ </tex-math></inline-formula>. The novelty of the construction is concentrated in five original structural theorems: (i) an unconditional cross-correlation modular-constancy theorem <inline-formula> <tex-math notation="LaTeX">$\nu \equiv 1\pmod q$ </tex-math></inline-formula> at <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> odd, <inline-formula> <tex-math notation="LaTeX">$d\ge 3$ </tex-math></inline-formula> with a sharp two-value pattern <inline-formula> <tex-math notation="LaTeX">$V_{2}-V_{1}{\,}\text {=}{\,}q^{(d-1)/2}$ </tex-math></inline-formula>, proving a 2-torsion distance ceiling; (ii) a Chinese-Remainder-Theorem (CRT) zero-count theorem <inline-formula> <tex-math notation="LaTeX">$\#\{f:\tau _{H,f}{\,}\text {=}{\,}0\}{\,}\text {=}{\,}(|\mathcal {F}|-1)/2$ </tex-math></inline-formula>; (iii) an explicit Clifford bridge <inline-formula> <tex-math notation="LaTeX">$U_{M}{\,}\text {=}{\,}H^{\otimes n}{\,}\mathrm {CZ}^{(M)}{\,}H^{\otimes n}$ </tex-math></inline-formula> that identifies <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(q,2)$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>; (iv) a paired column-rigidity equality <inline-formula> <tex-math notation="LaTeX">$d_{\min }{\,}\text {=}{\,}R_{P}+1$ </tex-math></inline-formula> exactly; and (v) a rigorous lower bound <inline-formula> <tex-math notation="LaTeX">$d_{\min }(\mathcal {Q}(7,2))\ge 8$ </tex-math></inline-formula> from an exhaustive Singer-Frobenius search. Via a Hamada-Smith specialisation we obtain the closed-form logical dimension <inline-formula> <tex-math notation="LaTeX">$k(p,d){\,}\text {=}{\,}\theta _{d}(p)-\binom {p+d-1}{d}-1$ </tex-math></inline-formula>, with the flagship <inline-formula> <tex-math notation="LaTeX">$d{\,}\text {=}{\,}2$ </tex-math></inline-formula> family <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(p,2){\,}\text {=}{\,}[[p^{2}+p+1,{\,}p(p+1)/2,{\,}d_{\min }]]_{p}$ </tex-math></inline-formula> attaining asymptotic rate <inline-formula> <tex-math notation="LaTeX">$\tfrac {1}{2}$ </tex-math></inline-formula>. For general odd primes <inline-formula> <tex-math notation="LaTeX">$p\ge 3$ </tex-math></inline-formula> the flagship distance is conjectured to satisfy <inline-formula> <tex-math notation="LaTeX">$d_{\min }(\mathcal {Q}(p,2)){\,}\text {=}{\,}p+1$ </tex-math></inline-formula>; this conjecture is analytically proven exact at <inline-formula> <tex-math notation="LaTeX">$p{\,}\text {=}{\,}5$ </tex-math></inline-formula> and exhaustively lower-bounded to <inline-formula> <tex-math notation="LaTeX">$d_{\min }\ge 8$ </tex-math></inline-formula> at <inline-formula> <tex-math notation="LaTeX">$p{\,}\text {=}{\,}7$ </tex-math></inline-formula>, and remains open for general <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>. Concrete instances include <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(3,2){\,}\text {=}{\,}[[{13,6,3}]]_{3}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(5,2){\,}\text {=}{\,}[[{31,15,6}]]_{5}$ </tex-math></inline-formula> (double-error correcting), <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(7,2){\,}\text {=}{\,}[[57,28,d{\ge }8]]_{7}$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(3,4){\,}\text {=}{\,}[[{121,105,3}]]_{3}$ </tex-math></inline-formula>. Monte-Carlo simulations over <inline-formula> <tex-math notation="LaTeX">$1.5{\times }10^{6}$ </tex-math></inline-formula> trials show that <inline-formula> <tex-math notation="LaTeX">$\mathcal {Q}(5,2)$ </tex-math></inline-formula> achieves a <inline-formula> <tex-math notation="LaTeX">$\sim 70\times $ </tex-math></inline-formula> per-logical-qudit improvement over Steane <inline-formula> <tex-math notation="LaTeX">$[[{7,1,3}]]_{2}$ </tex-math></inline-formula> at depolarising rate <inline-formula> <tex-math notation="LaTeX">$p_{\mathrm {ch}}{\,}\text {=}{\,}10^{-3}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, 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.