Admissible Line Complexes in Z2⁴

• The paper establishes that an admissible line complex in Z2⁴, comprised of exactly 16 affine lines, guarantees the discrete X‐ray transform is injective via a full rank incidence matrix.
• It details the decomposition into odd cycles with attached trees, identifying obstructions like even cycles and isolated vertices that affect injectivity.
• It enumerates admissible configurations, showing there are approximately 9.84×10^17 complexes, with applications in tomographic imaging and incidence geometry.

An admissible line complex in $\mathbb{Z}_{2}^{4}$ is a collection of affine lines in the 16-point vector space $V_4 = \mathbb{Z}_{2}^{4}$ such that the associated discrete X-ray (line) transform restricts to an injective linear map on functions from $V_4$. These structures are of fundamental interest both in combinatorial integral geometry, particularly as discrete models for tomographic data acquisition, and for their connections to incidence matrices and graph-theoretical properties of finite vector spaces (Dusad, 13 Dec 2025).

1. Definitions and Basic Properties

Let $V_4 = \mathbb{Z}_{2}^{4}$. The affine lines are unordered pairs $\{x, y\}$ of distinct elements of $V_4$. A line complex $\mathcal{C}$ is any subset of these lines and can be identified with an undirected, simple graph $G(\mathcal{C})$ on 16 vertices, with edges corresponding to the selected affine lines.

For $f: V_4 \to \mathbb{C}$, the discrete X-ray (or line) transform is defined by

$$Rf(\{x, y\}) = f(x) + f(y), \qquad \{x, y\} \in \mathcal{L},$$

where $\mathcal{L}$ denotes the collection of all lines.

The map $$R_{\mathcal{C}}: \mathbb{C}^{V_4} \to \mathbb{C}^{\mathcal{C}}$$ sends $f$ to the tuple of line sums $(Rf(L))_{L \in \mathcal{C}}$. The line complex $\mathcal{C}$ is termed admissible if $R_{\mathcal{C}}$ is injective, which is equivalent to the condition that the $|\mathcal{C}| \times 16$ incidence matrix $M_{\mathcal{C}}$ (with entries $m_{L,x}=1$ if $x \in L$, 0 otherwise) has full column rank 16. Hence, admissible complexes contain at least 16 lines (Dusad, 13 Dec 2025).

2. Structural Characterization: Odd-Cycle + Trees Decomposition

A fundamental structural insight is the characterization of obstructions to admissibility. If the graph $G(\mathcal{C})$ has an omitted vertex, a tree as a component, or an even-length cycle, then $R_{\mathcal{C}}$ is not injective. If these obstructions are absent, then each connected component must consist of a single odd cycle, possibly with trees attached at cycle vertices. Explicitly: each connected component contains exactly one odd cycle (length $\ell\in\{3,5,7,9,11,13,15\}$), with the remainder of the vertices forming attached trees.

The structural classification theorem is as follows:

Let $\mathcal{C}$ be a line complex in $V_n = \mathbb{Z}_{2}^{n}$. If (i) every vertex is incident to at least one edge of $\mathcal{C}$, (ii) $G(\mathcal{C})$ contains no even cycles, and (iii) no component is a tree, then $R_{\mathcal{C}}$ is injective and each component is an odd cycle with attached trees (Dusad, 13 Dec 2025).

In $V_4$, each admissible complex decomposes into such odd cycle plus tree components.

3. Enumeration of Admissible Line Complexes in $\mathbb{Z}_{2}^{4}$

Given the requirement that admissible complexes must yield an incidence matrix of rank 16, any admissible $\mathcal{C} \subset \mathcal{L}$ must contain exactly 16 lines. Each component contributes as many lines as it contains vertices, and the total number of component vertices $\sum_j \ell_j \leq 16$, where each $\ell_j$ is an odd integer $\ge 3$.

The enumeration involves:

• Listing all compositions of 16 into sums of odd integers $\ge 3$.
• For each composition, count the ways to select mutually disjoint vertex subsets and orderings for cycles.
• The number of simple cycles of length $\ell$ on labeled vertices is $\frac{(\ell-1)!}{2}$.
• Remaining (tree) vertices are distributed among the cycle vertices; if a cycle vertex receives $a$ attachments, there are $(a+1)^{a-1}$ labeled trees (Cayley's formula).

Summing over all decompositions and configurations gives the total: $$\#\{\text{admissible line complexes in }\mathbb{Z}_{2}^{4}\} = 984{,}014{,}621{,}487{,}058{,}560 = 9.84014621487058560 \times 10^{17}$$ (Dusad, 13 Dec 2025).

The one-cycle cases, listed by $\ell$, appear in the following table:

Cycle Length $\ell$	Number of Cycle Selections	Tree Attachment Ways	Total One-Cycle Contributions
3	560	$13^{11}$	$4.7287 \times 10^{17}$
5	43,680	$11^{9}$	$2.8816 \times 10^{17}$
7	...	$9^{7}$	$1.2382 \times 10^{17}$
9	...	$7^{5}$	$3.4824 \times 10^{16}$
11	...	$5^{3}$	$5.7133 \times 10^{15}$
13	...	$3^1$	$4.4635 \times 10^{14}$
15	...	$1^0$	$1.0461 \times 10^{13}$

Multi-cycle types are handled analogously.

4. Algorithmic Generalization to Higher Dimensions

The above enumeration extends to $V_n = \mathbb{Z}_2^n$, $N=2^n$.

Let $m = \ell_1+\cdots+\ell_r \le N$, $\ell_j$ odd $\ge 3$. For each partition:

• Disjoint subsets selected in $\binom{N}{\ell_1, \ldots, \ell_r, N-m}$ ways.
• Cycles chosen via $\prod_j \frac{(\ell_j-1)!}{2}$.
• Distribution of $N-m$ tree vertices among $m$ cycle sites in all compositions $(a_1, \ldots, a_m)$.
• Each tree-attachment contributes $\prod_{i=1}^m (a_i+1)^{a_i-1}$.

All admissible complexes are thus systematically enumerable via this algorithm.

This process is computationally feasible for $n\leq 5$, yielding, for $n=5$,

$$\#\{\text{admissible complexes in }\mathbb{Z}_2^5\} = 6.817 \times 10^{46}.$$

The admissible fraction drops rapidly as $n$ increases (for $n=5$, approximately $0.03\%$ of all possible $32$-edge subfamilies) (Dusad, 13 Dec 2025).

5. Concrete Examples and Graphical Interpretation

Explicit models in $V_4$ illustrate the cycle-plus-tree paradigm:

• Example 1 (Triangle plus pendant edge): Let $\{1,2,3\}$ be a 3-cycle, attach vertex 4 to vertex 1. The submatrix of the incidence matrix on these four vertices is invertible, and the structure is admissible. This construction generalizes to larger complexes by extending similar motifs.
• Example 2 (Disjoint 5-cycle): Five vertices $a,b,c,d,e$ in a 5-cycle (no attached trees), the incidence submatrix is circulant and invertible. The remaining 11 vertices may be attached as trees to the cycle vertices, in any of $(11+5)^{11}$ ways.

Each admissible complex in $V_4$ thus corresponds to a simple graph that is a forest of odd cycles with branching trees.

6. Connections to Incidence Geometry and Classical Line Complexes

The admissibility problem for line complexes in $V_4$ has deep ties with classical incidence geometry, notably in the context of projective 3-space over $\mathbb{F}_2$ (PG(3,2)). Admissible line complexes correspond to structures characterized algebraically via nonzero alternating bilinear forms (null polarities), yielding linear complexes of lines in PG(3,2) (Havlicek et al., 2013). These are classified up to projective equivalence into two types: the non-degenerate symplectic complex and the degenerate rank-2 complex with singular points. The vector space of $4\times4$ alternating matrices over $\mathbb{F}_2$ has $2^6-1=63$ nontrivial forms, accounting for all linear line complexes in this context.

For the projective Radon transform, minimal admissibility corresponds to certain 15-line complexes, classified combinatorially by avoidance of reguli in doubly ruled quadrics (DRQs). The unique (up to projective equivalence) minimal admissible complex avoids all three-line reguli from any DRQ, and there are precisely 30 such complexes in PG(3,2) (Feldman et al., 2017).

7. Applications and Further Directions

The admissible complexes explicitly model discrete sampling schemes in tomographic imaging—reconstructing functional data on finite vector spaces from sums over lines. The combinatorial, spectral, and algebraic classifications inform algorithmic approaches for higher-dimensional integral geometry. Moreover, connections to the rank and kernel structure of incidence matrices, and the restriction of the Radon transform to combinatorially structured subfamilies of lines, provide frameworks for further exploration in both discrete geometry and applied areas such as compressed sensing or error-correcting codes.

A plausible implication is that as $n$ grows, the structure of admissible line complexes becomes increasingly intricate, with the proportion of admissible subfamilies diminishing rapidly. This suggests combinatorial and algorithmic challenges in higher-dimensional settings and deeper connections to the spectral theory of graphs associated with $\mathbb{Z}_2^n$ (Dusad, 13 Dec 2025).