Conjecture on the 2-rank of the matrix $M_2$

Let $M_2$ be the incidence matrix defined in Subsection 3.2, and let $q$ be the parameter of the construction. The $2$ -rank of a matrix is its rank over the field with two elements.

Conjecture on the 2-rank of $M_2$ .

\operatorname{rank}_2(M_2)=q^4.

The preceding lower bound gives $\operatorname{rank}_2(M_2)\geq q^4-q^3-q^2+q$ , while computations using MAGMA yield the values $81$ , $625$ , and $2401$ for $q=3,5,7$ , respectively. The conjecture proposes the exact 2-rank suggested by these data.

References

Primary source

Lijun Ma, Changli Ma and Zihong Tian, “Strongly regular generalized partial geometries and associated LDPC codes”, arXiv:2503.14058 (2025).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-written argument claims a complete proof of the conjecture for all odd prime powers, but nobody has independently verified it.

The conjecture asserts that the binary rank of $M_2$ equals $q^4$ . The construction and conjecture appear in the 2025 paper by Lijun Ma, Changli Ma, and Zihong Tian.

Known results

The recorded lower bound is $\operatorname{rank}_2(M_2)\ge q^4-q^3-q^2+q$ .
MAGMA computations give ranks $81$ , $625$ , and $2401$ for $q=3,5,7$ , matching $q^4$ .

Posted attempt

The attempted proof identifies the columns with affine lines in $M_2(\mathbb F_q)$ having invertible directions, then uses additive characters over a characteristic-two extension to claim that every character lies in the column span. It claims a complete proof for every odd prime power, but the argument has not been independently verified.

Current status (as of August 2026): the conjecture has a complete but unverified posted proof claim; the published source records only the lower bound and confirming computations, so independent validation remains open.

Sources

arxiv.org

ProofThis solution needs a summarySee full solutionHide full solution

Let $q=p^e$ be any odd prime power and let

V=M_2(\mathbb F_q).

In the notation of Subsection 3.2, rows are indexed by $N\in V$ , and a hyperbolic-quadric column indexed by

B\in\mathrm{GL}_2(\mathbb F_q),\qquad C=C^{\mathsf T},

has incidence condition

NB+B^{\mathsf T}N^{\mathsf T}+C=0. \tag{1}

If $N_0$ satisfies (1), then $N$ satisfies it exactly when

(N-N_0)B

is skew-symmetric. Writing

J=\begin{pmatrix}0&1\\-1&0\end{pmatrix},

the complete column support is therefore the affine line

N_0+\mathbb F_qD, \qquad D=JB^{-1}\in\mathrm{GL}_2(\mathbb F_q). \tag{2}

Conversely, every invertible $D$ and every $N_0$ occur: choose

B=D^{-1}J,\qquad C=-(N_0B+B^{\mathsf T}N_0^{\mathsf T}).

Thus $M_2$ is the incidence matrix of all affine lines in $V$ with invertible matrix directions.

Choose a finite extension $E/\mathbb F_2$ containing a primitive $p$ -th root of unity $\zeta$ . For $A\in V$ , define

\chi_A(N) = \zeta^{\operatorname{Tr}_{\mathbb F_q/\mathbb F_p} (\operatorname{tr}(A^{\mathsf T}N))}.

Because $|V|=q^4$ is odd, these additive characters form an $E$ -basis of $E^V$ .

Let $W\subseteq E^V$ be the span of the column incidence vectors. Every translate of a line (2) is again a column, so $W$ is translation-invariant. For the line $L_D=\mathbb F_qD$ , its Fourier coefficient at $\chi_A$ is

\sum_{t\in\mathbb F_q} \zeta^{\operatorname{Tr}_{\mathbb F_q/\mathbb F_p} (t\operatorname{tr}(A^{\mathsf T}D))} = \begin{cases} q=1\in E,&\operatorname{tr}(A^{\mathsf T}D)=0,\\ 0,&\operatorname{tr}(A^{\mathsf T}D)\neq0. \end{cases} \tag{3}

Character orthogonality and translation invariance imply

\chi_A\in W

whenever there exists an invertible $D$ satisfying

\operatorname{tr}(A^{\mathsf T}D)=0. \tag{4}

Such a $D$ exists for every $A$ . If $A=0$ , take $D=I_2$ . If $\operatorname{rank}A=1$ , write

A=UE_{11}V,\qquad U,V\in\mathrm{GL}_2(\mathbb F_q),

and set

D=U^{-\mathsf T} \begin{pmatrix}0&1\\1&0\end{pmatrix} V^{-\mathsf T}.

If $\operatorname{rank}A=2$ , write $A=UI_2V$ and set

D=U^{-\mathsf T} \begin{pmatrix}1&0\\0&-1\end{pmatrix} V^{-\mathsf T}.

In both cases $D$ is invertible and satisfies (4).

Therefore $W$ contains all $q^4$ additive characters, so

W=E^V.

Scalar extension preserves the rank of a matrix over $\mathbb F_2$ . Consequently,

\boxed{\operatorname{rank}_{\mathbb F_2}(M_2)=q^4}

for every odd prime power $q$ , as conjectured.