bibkey: kulhandjian2026singer authors: M. Kulhandjian; L. Hanzo year: 2026 title: “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 claim: “Conjecture 24 asserts that Proposition 23 holds for every d >= 2: for q = 2 the cross-correlation u = M_Q tau_H of the code Q(2,d) is the all-ones vector, every Z_i Z_j is a non-stabilizer element of the centralizer, and the minimum distance is 2.” strata_touched:

• D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling
• D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound license: citation-only triage: anchor

Singer-difference-set qudit stabilizer codes

M. Kulhandjian and L. Hanzo, arXiv:2610.02392v1 (2026-10-01), quant-ph and cs.IT; comment: accepted for publication in IEEE Open Journal of the Communications Society. Source file Paper_OJCOMS.tex. Proposition 23 and Conjecture 24 are in Section VII.E; the construction is the Algorithm of Section V and Appendix A.

The source sentences are quoted verbatim below.

\begin{proposition}[$q=2$ distance ceiling; original, numerical]\label{prop:q2-ceiling} For every $d\in \{2,3,4,5,6\}$, the cross-correlation u=M_{Q}\boldsymbol\tau_{H}\in\F 2^{n} of $\mathcal{Q}(2,d)$ satisfies $u\equiv 1(\mathrm{mod}2)$ (verified by \texttt{verify_q2_ceiling.py}). Hence $H_x=\mathrm{circ}(u)$ is the all-ones circulant mod 2, and for any pair of distinct positions $\{i,j\}$ the Pauli $\mathbf{a}=e_i+e_j$ satisfies $\mathbf{a}{H}_x^{\top }=u_i+u_j=0(\mathrm{mod}2)$, putting $Z_i{Z}_j$ in the centralizer of $\mathcal{Q}(2,d)$. At $(2,2),(2,3),(2,4)$ exhaustive search certifies that $Z_0{Z}_1$ is \emph{not} in $\mathrm{rowspan}(H)$, so it is a non-stabilizer weight-$2$ logical and $d_{\min }(\mathcal{Q}(2,d))=2$ exactly. \end{proposition}

\begin{conjecture}[General-$d$ $q=2$ ceiling]\label{conj:q2-ceiling-general} Proposition~\ref{prop:q2-ceiling} holds for every $d\ge 2$. \end{conjecture}

The list of open problems in the conclusions repeats it:

\item \emph{General-$d$ $q=2$ ceiling.} Settle Conjecture~\ref{conj:q2-ceiling-general} analytically; numerically confirmed at $d\le 6$.

The construction at $q=2$ (Appendix A, Steps 2–4, and the Algorithm output), verbatim:

Define \boldsymbol\tau_{H}\in\F q^{n} by $({\mathbf{\tau }}_H{)}_i=1$ if $T[i]=0$, else $0$, for $i=0,\dots ,n-1$.

Set $\xi =q+1$ if $q$ is even, $\xi =2$ if $q$ is odd. Define \boldsymbol\tau_{Q}\in\F q^{n} by $({\mathbf{\tau }}_Q{)}_i=1$ if $T[(\xi i)\mathrm{mod}(q^{d+1}-1)]=0$, else $0$.

$M_Q=\mathrm{circ}({\mathbf{\tau }}_Q)$, the $n\times n$ circulant matrix with first column ${\mathbf{\tau }}_Q$ and subsequent columns cyclic-shifts thereof: $(M_Q{)}_{i,j}=({\mathbf{\tau }}_Q{)}_{(i-j)\mathrm{mod}n}$.

Compute the centralizer \mathcal C=\{(\mathbf a,\mathbf b)\in\F q^{2n}\colon \mathbf a H_{x}^{\top}=\mathbf b H_{z}^{\top}\}

d_{\min}=\min\{\wt(\mathbf v)\colon \mathbf v\in \mathcal C\setminus\mathcal S\}.

Here T[i]=\Tr_{K/F}(\alpha^i) for a primitive $\alpha \in {\mathbb{F}}_{q^{d+1}}$, $n=(q^{d+1}-1)/(q-1)$, $A=\mathrm{circ}({\mathbf{\tau }}_H)$ is the Singer incidence matrix, $H=(H_z\mid H_x)=(A\mid M_{Q}A)$ and $\mathcal{S}$ is the row space of $H$.

Verified locator

DOI: 10.48550/arXiv.2610.02392. Canonical source URL: https://arxiv.org/abs/2610.02392v1. Proposition 23 (prop:q2-ceiling) and Conjecture 24 (conj:q2-ceiling-general) are in Section VII.E; the open-problem list is in Section X; the construction is in Appendix A.

Conjecture 25, Section VII, page 19 (verbatim source TeX):

For prime $p$ odd, $d_{\min }(\mathcal{Q}(p,2))\le p+1$.

The trace-plane definition, Section III, page 7 (verbatim source TeX):

Its first column ${\mathbf{\tau }}_H$ has $({\mathbf{\tau }}_H{)}_i=1$ iff \Tr_{K/F}(\alpha^i)=0.

The odd-prime construction takes $K={\mathbb{F}}_{p^3}$ and $n=p^2+p+1$, with ${\tau }_H(i)=[\mathrm{Tr}({\alpha }^i)=0]$ and ${\tau }_Q(i)=[\mathrm{Tr}({\alpha }^{2i})=0]$. The trace-plane description is on page 7 and the algorithm in Appendix A on pages 27–28.

The Appendix A worked example at $p=3$, using $X^3+2X^2+X+1$ and $\alpha =X$, prints $D=\{2,3,6,8\}$ and $Q=\{0,7,8,11\}$. The defining trace formulas instead give $D=\{0,7,8,11\}$ and $Q=\{0,4,10,12\}$, with $Q=2^{-1}D$ modulo 13. These corrected supports are obtained by exact arithmetic in the stated cubic polynomial quotient; the proof uses the defining trace formulas.