Symmetry-Induced Optimal Recursion Depth in Projective Resolvable Designs

Abstract

Recursive constructions derived from finite projective geometries generate scalable families of resolvable block designs exhibiting strong algebraic regularity and intrinsic symmetry. In this work, we investigate the structural optimization of recursion depth in such constructions and demonstrate that the combinatorial growth of recursive chains is governed by a quadratic scaling law originating from the asymptotic expansion of Gaussian binomial coefficients. We show that the resulting exponent is strictly concave, which guarantees the existence and uniqueness of an optimal recursion depth. This optimum occurs at the midpoint of the projective dimension and reflects the dual symmetry of the lattice of projective subspaces. To analyze this behavior, we introduce a normalized objective function that compares recursion depths and reveals a unique maximum corresponding to the midpoint of the projective dimension. Theoretical analysis is supported by exact enumeration and asymptotic validation, confirming that the optimal depth is robust to lower-order perturbations and remains invariant under the natural duality of projective geometry. The proposed framework establishes a direct connection between symmetry properties of discrete geometric structures and optimality in nonlinear discrete systems. These results provide a unified perspective on recursive design constructions, revealing that symmetry not only governs combinatorial structure but also induces a mathematically inevitable optimal configuration. The approach opens new directions for studying symmetry-induced optimality in combinatorial geometry, discrete optimization, and related nonlinear mathematical models.

1. Introduction

Symmetry is a fundamental organizing principle in mathematics, often underlying both structural regularity and the emergence of optimal configurations. In discrete settings, this principle is typically encoded through algebraic and combinatorial structures whose internal balance leads naturally to extremal behavior. Finite projective geometries provide a particularly rich framework in this regard: duality and dimensional complementarity generate highly regular lattices of subspaces, and these symmetries strongly constrain associated combinatorial constructions.

One important manifestation of this structure is found in block designs. These incidence configurations translate geometric regularity into combinatorial schemes with controlled balance and redundancy, making them central objects in both combinatorics and statistical experimentation. In particular, balanced incomplete block designs (BIBDs) and resolvable designs play a key role in experimental design, where they ensure controlled replication and pairwise concurrence [1,2]. Finite projective geometries constitute a systematic source of such designs, yielding families with strong algebraic structure and recursive construction mechanisms.

Classical studies of projective designs have focused primarily on existence, symmetry, and combinatorial balance. In contrast, many modern applications require an additional level of flexibility: constructions must accommodate constraints on block size, replication, and computational complexity, while optimal design theory provides well-established criteria such as D-, A-, and E-optimality [3,4], these operate at the level of statistical inference and do not directly address structural parameters arising in recursive combinatorial constructions.

At the same time, multi-criteria optimization has developed a range of tools for aggregating competing objectives. Among these, desirability functions, originally proposed by Harrington and later formalized by Derringer and Suich [5], offer a flexible way to combine heterogeneous criteria into a single scalar measure. Originally introduced in quality engineering, they have since been adapted to robust optimization and multi-response modeling [6,7,8,9]. More broadly, advances in multi-objective optimization—including Pareto methods, evolutionary approaches, and scalarization techniques [10,11]—have reinforced the role of aggregated criteria in guiding complex design choices.

Related questions have also appeared in the study of recursive combinatorial constructions. Ref. [12] investigated recursive methods for orthogonal arrays, while ref. [13] explored links between projective geometries and covering arrays. Recent work on subspace configurations and decompositions in finite incidence structures further illustrates the continuing relevance of projective and polar spaces in design theory [14]. In parallel, Gaussian binomial coefficients have been studied extensively from both algebraic and enumerative viewpoints; in particular, established strict unimodality results that highlight the strong symmetry present in these coefficients. However, the question of selecting an optimal recursion depth in projective recursive constructions does not appear to have been addressed explicitly.

This gap motivates the present work. Our aim is to understand how the intrinsic symmetry of finite projective geometries influences the structural growth of recursive constructions of resolvable block designs and, more specifically, how it determines a preferred recursion depth. We study the evolution of the number of generated blocks as a function of depth and show that this growth is governed by Gaussian binomial coefficients whose asymptotic form leads to a quadratic exponent reflecting projective duality.

The resulting scaling law shows that the dominant exponent is strictly concave. As a consequence, the recursive family admits a unique optimal depth in the continuous model, with the usual discrete rounding in the integer setting. This optimum occurs at the midpoint of the projective dimension, where the effects of combinatorial expansion and dimensional contraction are balanced. To make this interpretation operational, we introduce a normalized structural desirability function that allows recursion depths to be compared within a common framework.

The main contributions of the paper are threefold. First, we derive the quadratic scaling law governing the number of blocks in the recursive construction and relate its form to the symmetry of Gaussian binomial coefficients. Second, we prove that this law is strictly concave and identify the corresponding optimal recursion depth explicitly in terms of the ambient dimension. Third, we introduce a desirability-based interpretation of this optimum and support the theoretical analysis with exact enumeration and asymptotic comparisons.

These results show that the optimal structural regime is not imposed externally by an auxiliary criterion, but is already encoded in the internal symmetry of the projective construction. In this sense, the paper connects geometric duality, combinatorial growth, and parameter selection within a single framework. The analysis also suggests that similar symmetry-induced optimality phenomena may arise in other algebraic incidence structures, including symplectic or unitary geometries.

To illustrate the theory, we examine the case $PG(7,2)$ in detail and compare exact block counts with the asymptotic prediction. The computations confirm that the bounded $O\left(1\right)$ correction term does not alter the location of the optimum. We also compare the resulting recursive family with more classical projective design constructions, showing that the midpoint regime occupies a distinct part of the design space characterized by very large block multiplicities and reduced block size.

This paper is organized as follows: Section 2 studies the structural growth of the recursive construction and derives the quadratic scaling law from projective-geometric considerations. Section 3 analyzes this law and establishes the existence and location of the structural optimum. Section 4 introduces the normalized desirability function and examines its concavity and symmetry properties. Section 5 returns to the recursive projective construction and shows how the quadratic exponent arises from the factorization of Gaussian binomial coefficients. Section 6 presents the computational analysis, including exact enumeration, visualization, and comparison with classical design families. Section 7 concludes with a discussion of implications and possible extensions.

2. Structural Growth of the Recursive Construction

Finite projective geometries form lattices of subspaces with strong symmetry properties, where duality and dimensional complementarity lead to a high degree of combinatorial regularity. The recursive construction considered later in Section 4 builds on this structure by iteratively nesting linear sub-varieties in $PG(m,p)$.

A block at depth n can therefore be described by a chain of inclusions

$$V\left(i_1\right)\supset V(i_1,i_2)\supset \cdots \supset V(i_1,\dots ,i_n),$$

which may be viewed as a partial flag in $PG(m,p)$.

In particular, the growth of the construction depends on the number of such nested chains, rather than only on the terminal subspaces.

It is well known that the number of $m_{j+1}$-dimensional sub-varieties contained in a fixed $m_j$-dimensional sub-variety is

$$N(m,m_j,m_{j+1})=1+p+\cdots +p^j.$$

The total multiplicative contribution over n levels is therefore

$${\alpha }_n={\displaystyle \prod _{i=1}^{n-1}}(1+p+\cdots +p^i).$$

To estimate this growth, we examine the asymptotic behavior of ${\alpha }_n$.

Proposition 1.

For fixed p and large n,

$${log}_p{\alpha }_n={\displaystyle \frac{n(n-1)}{2}}+O\left(n\right),$$

and therefore

$${\alpha }_n=p^{{\displaystyle \frac{n^2-n}{2}}+O\left(n\right)}.$$

Proof. 

Using the identity

$$1+p+\cdots +p^i={\displaystyle \frac{p^{i+1}-1}{p-1}}=p^i·{\displaystyle \frac{p-p^{-i}}{p-1}}.$$

Taking logarithms in base p, we obtain

$${log}_p(1+p+\cdots +p^i)=i+{log}_p\left({\displaystyle \frac{p-p^{-i}}{p-1}}\right).$$

The second term remains bounded for fixed p, so

$${log}_p{\alpha }_n={\displaystyle \sum _{i=1}^{n-1}}i+O\left(n\right)={\displaystyle \frac{(n-1)n}{2}}+O\left(n\right)={\displaystyle \frac{n(n-1)}{2}}+O\left(n\right).$$

Exponentiating completes the proof. □

The quantity ${\alpha }_n$ counts the number of admissible nested chains of length n in $PG(m,p)$. In the reduced design ${\mathcal{Q}}_n^{*}$, blocks are indexed by terminal $(m-n)$-dimensional subspaces rather than by individual chains, while multiple chains may lead to the same terminal subspace, each block corresponds uniquely to such a subspace. Consequently, the number of blocks $b_F^{\left(n\right)}$ coincides with the number of $(m-n)$-dimensional subspaces of $PG(m,p)$.

The number of r-dimensional subspaces of a vector space of dimension $m+1$ over ${\mathbb{F}}_p$ is given by the Gaussian binomial coefficient

$${\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{r}}\right)}_p.$$

Setting $r=m-n+1$ yields

$$b_F^{\left(n\right)}={\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m-n+1}}\right)}_p.$$

For fixed p and large parameters, this coefficient satisfies,

$${log}_p{\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m-n+1}}\right)}_p=nm-n^2+n+O\left(1\right).$$

The Gaussian binomial coefficient inherits a fundamental symmetry

$${\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{k}}\right)}_p={\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m+1-k}}\right)}_p,$$

reflecting the duality between complementary projective subspaces. This symmetry is reflected in the quadratic form of the logarithmic growth.

Theorem 1

(Structural Scaling Law). For fixed p and m,

$${log}_p{b}_F^{\left(n\right)}=nm-n^2+n+O\left(1\right),$$

and therefore

$$b_F^{\left(n\right)}=p^{nm-n^2+n+O\left(1\right)}.$$

At depth n, the block size satisfies

$$k_n^{*}=p^{m-n}.$$

As n increases, the block size decreases exponentially, while the number of blocks follows a quadratic growth pattern in the exponent.

The dominant combinatorial behavior is captured by the function

$$g\left(n\right)=nm-n^2+n.$$

(1)

We next examine a key structural property of this function.

Proposition 2

(Strict concavity of the scaling law). The function $g\left(n\right)$ defined in (1) is strictly concave on $\mathbb{R}$.

Proof. 

We have

$$g^{\prime }\left(n\right)=m+1-2n,g^{″}\left(n\right)=-2.$$

Since $g^{″}\left(n\right)=-2<0$, the function is strictly concave on $\mathbb{R}$. This also implies concavity on the integer lattice.

This ensures that the function is unimodal and uniqueness of the optimum. □

Corollary 1

(Existence and Uniqueness of Optimal Depth). The function $g\left(n\right)$ admits a unique maximizing depth in the admissible discrete interval $0\le n\le m$.

If m is odd, the maximum is attained uniquely at

$$n^{*}={\displaystyle \frac{m+1}{2}}.$$

If m is even, the maximum is attained at the two adjacent integers

$$n^{*}={\displaystyle \frac{m}{2}}and{n}^{*}={\displaystyle \frac{m+2}{2}},$$

corresponding, respectively, to

$$⌊{\displaystyle \frac{m+1}{2}}⌋and⌈{\displaystyle \frac{m+1}{2}}⌉.$$

Proof. 

The maximum is obtained when $\Delta g\left(n\right)=0$, which gives

$$m+1-2n=0.$$

Strict concavity ensures uniqueness up to the unavoidable discrete rounding. □

The scaling law (1) captures the dominant combinatorial behavior. It reflects the balance between dimensional reduction and the growth in the number of admissible configurations, a feature that will play a central role in the analysis that follows.

3. Quadratic Scaling Law and Structural Optimality

Section 2 established the asymptotic scaling law through the exponent $g\left(n\right)$ defined in (1), which governs the combinatorial growth of the recursive construction.

The quadratic form of $g\left(n\right)$ encodes the interaction between two opposing mechanisms inherent in the construction: the increase in the number of admissible configurations induced by deeper nesting, and the reduction in block size caused by successive dimensional contraction.

As shown in Proposition 2, the function $g\left(n\right)$ is strictly concave. This property has a direct structural consequence: the growth profile is unimodal, and the maximum is unique.

From Corollary 1, the optimal recursion depth is given by

$$n^{*}={\displaystyle \frac{m+1}{2}},$$

up to the usual integer adjustment when m is even.

This midpoint has a natural interpretation. For small values of n, the combinatorial expansion dominates, leading to a rapid increase in the number of configurations. For larger values of n, the progressive reduction in dimension limits further growth. The optimal depth corresponds to the point where these two effects balance.

In this sense, the scaling law identifies a preferred structural regime within the lattice of projective subspaces. The location of the optimum reflects the symmetry of the Gaussian binomial coefficients, which relate subspaces of complementary dimensions in $PG(m,p)$.

Thus, the recursive family ${\mathcal{Q}}_n^{*}$ achieves its maximal combinatorial richness at half-depth, where the underlying geometric symmetry is most strongly expressed.

4. Symmetry-Induced Desirability Surface

To compare recursion depths, we introduce a normalized desirability function that aggregates the relevant structural quantities into a single measure. This function provides a convenient way to identify depths that balance growth and structural constraints.

We define the desirability surface as a mapping

$$D:\mathcal{K}\to [0,1],$$

where $D\left(n\right)$ measures the relative suitability of recursion depth n.

The dominant exponent (1) governs the combinatorial growth of the construction in $PG(m,p)$.

From Section 3, the continuous maximizer of g is

$$n^{*}={\displaystyle \frac{m+1}{2}},$$

with corresponding value

$$g\left(n^{*}\right)={\displaystyle \frac{{(m+1)}^2}{4}}.$$

In the discrete setting, admissible depths are integers in $[0,m]$. When m is even, the maximum of $g\left(n\right)$ is attained at the two adjacent integers

$$⌊{\displaystyle \frac{m+1}{2}}⌋\mathrm{and}⌈{\displaystyle \frac{m+1}{2}}⌉.$$

For normalization, we use the continuous value $g\left(n^{*}\right)$, which avoids dependence on discrete rounding.

We define the normalized desirability function

$$D(p,m,n)={\displaystyle \frac{g\left(n\right)}{g\left(n^{*}\right)}}={\displaystyle \frac{nm-n^2+n}{\frac{{(m+1)}^2}{4}}}.$$

Equivalently,

$$D(p,m,n)={\displaystyle \frac{4((m+1)n-n^2)}{{(m+1)}^2}}.$$

By construction,

$$0\le D(p,m,n)\le 1,D(p,m,n^{*})=1.$$

In the discrete setting, the maximal integer depths attain values close to 1 as m increases.

Theorem 2

(Structure of the desirability surface). For fixed $(p,m)$, the function $n\mapsto D(p,m,n)$ is strictly concave in the discrete sense and attains its maximum (up to integer rounding) at

$$n^{*}={\displaystyle \frac{m+1}{2}}.$$

Proof. 

Since $D(p,m,n)$ is obtained from $g\left(n\right)$ by multiplication by the positive constant $4/{(m+1)}^2$, concavity is preserved. From Section 2,

$${\Delta }^{2}g\left(n\right)=-2<0,$$

which gives

$${\Delta }^{2}D(p,m,n)={\displaystyle \frac{4}{{(m+1)}^2}}{\Delta }^{2}g\left(n\right)=-{\displaystyle \frac{8}{{(m+1)}^2}}<0.$$

Thus, D is strictly concave. The maximizer coincides with that of g, namely $n^{*}={\displaystyle \frac{m+1}{2}}$. □

The symmetry of the desirability function follows from the identity

$${\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{k}}\right)}_p={\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m+1-k}}\right)}_p,$$

which reflects the duality between complementary projective subspaces. This symmetry is mirrored in the quadratic form of $g\left(n\right)$ and implies that the maximum occurs at the midpoint.

At depth

$$n^{*}={\displaystyle \frac{m+1}{2}},$$

the contributions of complementary dimensions are balanced. This corresponds to the point where the number of admissible configurations is maximized relative to the normalization.

As a result, the family ${\mathcal{Q}}_n^{*}$ attains its highest structural efficiency at half-depth. Strict concavity ensures that this optimum is unique, while symmetry explains its location.

Example 1

(Illustrative Example). We illustrate the structural scaling law and midpoint optimality in the concrete case of the projective geometry $PG(7,2)$.

For $m=7$ and $p=2$, the dominant combinatorial exponent governing the number of blocks at recursion depth n is given by (1).

Evaluating $g\left(n\right)$ for integer depths $1\le n\le 7$ yields

$$\begin{array}{cccccccc}n& 1& 2& 3& 4& 5& 6& 7\\ g\left(n\right)& 7& 12& 15& 16& 15& 12& 7\end{array}$$

The maximum occurs at

$$n^{*}={\displaystyle \frac{7+1}{2}}=4,$$

with a maximal value of $g\left(4\right)=16$.

The corresponding normalized desirability values

$$D\left(n\right)={\displaystyle \frac{g\left(n\right)}{g\left(4\right)}}$$

are

$$\begin{array}{cccccccc}n& 1& 2& 3& 4& 5& 6& 7\\ D\left(n\right)& 0.4375& 0.75& 0.9375& 1& 0.9375& 0.75& 0.4375\end{array}$$

Several structural features are immediately visible:

• The sequence is symmetric with respect to $n\leftrightarrow 8-n$, reflecting the Gaussian identity

  $${\left({\displaystyle \genfrac{}{}{0pt}{}{8}{k}}\right)}_2={\left({\displaystyle \genfrac{}{}{0pt}{}{8}{8-k}}\right)}_2.$$
• The quadratic profile is strictly concave, with a unique maximum at the midpoint $n=4$.
• The symmetry of the exponent directly mirrors the duality between k-dimensional and $(m+1-k)$-dimensional subspaces in $PG(7,2)$.

This example concretely illustrates the general structural phenomenon established in Section 2, Section 3 and Section 4: recursive contraction achieves maximal combinatorial richness at half-depth, where the dual symmetry of the projective lattice is realized.

5. Recursive Projective Construction and Structural Factorization

The family of reduced designs ${\mathcal{Q}}_n^{*}$ is obtained by successive nesting of linear sub-varieties in the projective geometry $PG(m,p)$. We show that this construction leads naturally to the quadratic exponent (1).

Let $PG(m,p)$ denote the m-dimensional projective space over ${\mathbb{F}}_p$. A projective subspace of dimension r corresponds to a vector subspace of dimension $r+1$ in ${\mathbb{F}}_p^{m+1}$. The number of such subspaces is given by the Gaussian binomial coefficient

$${\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{r+1}}\right)}_p={\displaystyle \prod _{j=0}^r}{\displaystyle \frac{p^{m+1-j}-1}{p^{r+1-j}-1}}.$$

The recursive construction proceeds by dimensional contraction. At depth n, we consider chains of subspaces

$$V_m\supset V_{m-1}\supset \cdots \supset V_{m-n},$$

where each $V_{m-j}$ has dimension $m-j$.

The number of admissible subspaces at each step is

$$N(m_j,m_{j+1})={\displaystyle \frac{p^{m_j+1}-1}{p^{m_{j+1}+1}-1}}=1+p+\cdots +p^j.$$

Hence, the number of chains of length n is

$${\alpha }_n={\displaystyle \prod _{i=1}^{n-1}}(1+p+\cdots +p^i).$$

We now relate this construction to the Gaussian binomial coefficient.

Theorem 3.

The number of $(m-n)$-dimensional projective subspaces of $PG(m,p)$ satisfies

$${\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m-n+1}}\right)}_p={\displaystyle \prod _{i=1}^n}{\displaystyle \frac{p^{m-i+2}-1}{p^i-1}},$$

and its dominant exponent is

$$g\left(n\right)=nm-n^2+n.$$

Proof. 

Setting $r=m-n$ in the definition of the Gaussian binomial coefficient gives

$${\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m-n+1}}\right)}_p={\displaystyle \prod _{j=0}^{m-n}}{\displaystyle \frac{p^{m+1-j}-1}{p^{m-n+1-j}-1}}.$$

Reindexing with $i=m-j$ yields

$${\displaystyle \prod _{i=1}^n}{\displaystyle \frac{p^{m-i+2}-1}{p^i-1}}.$$

For fixed p and large parameters, the dominant terms are $p^{m-i+2}$ in the numerator and $p^i$ in the denominator. Taking logarithms base p gives

$${\displaystyle \sum _{i=1}^n}(m+2-2i).$$

Evaluating the sum,

$${\displaystyle \sum _{i=1}^n}(m+2-2i)=n(m+2)-n(n+1)=nm-n^2+n.$$

Thus,

$${log}_p{\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m-n+1}}\right)}_p=nm-n^2+n+O\left(1\right),$$

where the $O\left(1\right)$ term is bounded for fixed p. □

This shows that the recursive construction reproduces the dominant asymptotic behavior of the Gaussian binomial coefficient.

At depth n, each block corresponds to a subspace of dimension $m-n$, so its size is

$$k_n^{*}=p^{m-n}.$$

Thus, block size decreases exponentially with n, while the number of configurations grows according to the quadratic exponent.

Combining these observations gives

$${log}_p{b}_F^{\left(n\right)}=nm-n^2+n+O\left(1\right),$$

which matches the scaling law derived earlier.

The construction therefore traces a path through the lattice of projective subspaces. The optimal depth

$$n^{*}={\displaystyle \frac{m+1}{2}},$$

corresponds to the maximum of this exponent.

In summary, the recursive construction, the Gaussian factorization, and the concavity of $g\left(n\right)$ are consistent with one another and lead to the same optimal depth.

6. Computational Analysis and Comparative Evaluation

The theoretical framework developed in Section 2, Section 3, Section 4 and Section 5 establishes that the recursive family ${\mathcal{Q}}_n^{*}$ exhibits a strictly concave quadratic growth exponent (1), with a unique optimal recursion depth at $n^{*}=(m+1)/2$. In this section, we provide a comprehensive computational investigation that: (i) validates the asymptotic approximations against exact combinatorial counts; (ii) visualizes the desirability surface across parameter ranges; (iii) compares the structural properties of the recursive family with classical design constructions; and (iv) discusses the practical implications of the trade-off between block size and combinatorial richness.

6.1. Exact Enumeration and Asymptotic Validation

We begin by examining the exact behavior of the recursive construction for computationally tractable parameter values. For a fixed prime power p and ambient dimension m, the exact number of blocks at recursion depth n is given by the Gaussian binomial coefficient:

$$b_F^{\left(n\right)}={\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{m-n+1}}\right)}_p={\displaystyle \prod _{j=0}^{m-n}}{\displaystyle \frac{p^{m+1-j}-1}{p^{m-n+1-j}-1}}.$$

Our asymptotic analysis predicts that ${log}_p{b}_F^{\left(n\right)}=g\left(n\right)+O\left(1\right)$, where $g\left(n\right)=nm-n^2+n$. To assess the quality of this approximation, we computed exact values for the geometry PG(7, 2) (m = 7, p = 2) and compared them with the asymptotic prediction. The results are summarized in

Table 1. Exact Enumeration for $PG(7,2)$.

Depth n	Exact ${\mathit{b}}_{\mathit{F}}^{\left(\mathit{n}\right)}$	${log}_2{\mathit{b}}_{\mathit{F}}^{\left(\mathit{n}\right)}$	$\mathit{g}\left(\mathit{n}\right)$	Residual
1	255	7.99	7	0.99
2	10,795	13.40	12	1.40
3	97,155	16.57	15	1.57
4	200,787	17.62	16	1.62
5	97,155	16.57	15	1.57
6	10,795	13.40	12	1.40
7	255	7.99	7	0.99

.

Several important observations emerge from this enumeration that illuminate the relationship between the exact combinatorial structure and its asymptotic approximation:

1.

Perfect symmetry: The exact counts satisfy $b_F^{\left(n\right)}=b_F^{(8-n)}$, reflecting the duality relation ${\left({\displaystyle \genfrac{}{}{0pt}{}{8}{k}}\right)}_2={\left({\displaystyle \genfrac{}{}{0pt}{}{8}{8-k}}\right)}_2$. This symmetry is perfectly captured by the quadratic exponent $g\left(n\right)$, which satisfies $g\left(n\right)=g(8-n)$.

2.

Peak location: Both the exact counts and the asymptotic exponent achieve their maximum at $n=4$, with $b_F^{\left(4\right)}=200,787$ blocks and $g\left(4\right)=16$. This confirms that for $PG(7,2)$, the optimal recursion depth is indeed $n^{*}=4=(m+1)/2$.

3.

Residual pattern: The residuals ${log}_p{b}_F^{\left(n\right)}-g\left(n\right)$ are bounded and exhibit a symmetric pattern, increasing from 0.99 at the boundaries to 1.62 at the center. This confirms the theoretical prediction that the $O\left(1\right)$ term is bounded, while also revealing that it varies smoothly across the depth range, with maximum at the peak.

4.

Asymptotic accuracy: The maximum residual of 1.62 (at $n=4$) represents less than 10% of the ${log}_2{b}_F^{\left(4\right)}$ value of 17.62, demonstrating that the quadratic approximation captures the essential behavior with high fidelity even for moderate m. The relative error $\epsilon =|{log}_2{b}_F^{\left(n\right)}-g\left(n\right)|/{log}_2{b}_F^{\left(n\right)}$ ranges from 0.12 at the boundaries to 0.09 at the center, indicating excellent agreement.

To visualize the quality of the asymptotic approximation across a range of parameters and to demonstrate how the peak location aligns with increasing m, we computed the relative error $\epsilon \left(n\right)=|{log}_p{b}_F^{\left(n\right)}-g\left(n\right)|/{log}_p{b}_F^{\left(n\right)}$ for multiple geometries. Figure 1 displays these results for $p=2$ and varying m.

The figure confirms that for all $m\ge 5$, the quadratic approximation captures both the essential behavior and the peak location with high fidelity. For $m=5$, the peak occurs at $n=3=(5+1)/2$; for $m=7$, at $n=4=(7+1)/2$; for $m=9$, at $n=5=(9+1)/2$; and for $m=11$, at $n=6=(11+1)/2$. This perfect alignment validates the asymptotic analysis as a reliable guide for structural optimization across all dimensions.

The key insight is that the $O\left(1\right)$ term, while bounded and systematically varying, does not shift the location of the maximum. The optimal recursion depth is robustly given by $n^{*}=(m+1)/2$, as predicted by the asymptotic theory.

6.2. Visualization of the Structural Desirability Surface

The normalized desirability surface $D(p,m,n)=4((m+1)n-n^2)/{(m+1)}^2$ provides a scale-invariant measure of structural efficiency. We now examine its behavior across the continuous relaxation of the depth parameter and for integer depths.

For a fixed ambient dimension, say $m=50$, Figure 2 displays the desirability surface as a function of n.

The key features illustrated are:

• Strict concavity: The surface exhibits a single peak with no secondary optima, confirming the theoretical prediction.
• Symmetry: The surface is symmetric about $n^{*}=(m+1)/2$, reflecting the duality of the underlying subspace lattice.
• Integer approximation: For odd m, the optimal integer depth is uniquely $\lfloor (m+1)/2\rfloor =\lceil (m+1)/2\rceil$. For even m, the two adjacent integers $m/2$ and $(m+2)/2$ achieve desirability values that approach 1 as m increases, with the gap $1-D(m/2)$ decreasing as $O(1/m^2)$.

To illustrate the behavior across different geometries,

Table 2. Optimal Depths and Desirability for Various m.

m	${\mathit{n}}^{*}$ (Continuous)	Optimal Integer n	D at Optimal	$\mathit{g}\left({\mathit{n}}^{*}\right)$	${log}_2{\mathit{b}}_{\mathit{F}}^{\left({\mathit{n}}^{*}\right)}$ (Exact)
3	2.0	2	1.0000	4	4.39
4	2.5	2, 3	0.9600	6	6.68, 7.22
5	3.0	3	1.0000	9	9.88
6	3.5	3, 4	0.9796	12	12.91, 13.78
7	4.0	4	1.0000	16	17.62
8	4.5	4, 5	0.9877	20	20.97, 22.89
9	5.0	5	1.0000	25	27.51
10	5.5	5, 6	0.9917	30	31.78, 34.17

presents optimal depths and corresponding desirability values for a range of m values with $p=2$.

This table reveals an important practical insight: for even m, both adjacent integer depths achieve essentially the same desirability, and the difference from the continuous optimum becomes negligible for moderate m. Practitioners can therefore select either depth without meaningful loss of structural efficiency.

6.3. Comparative Structural Analysis

We compare the recursive family ${\mathcal{Q}}_n^{*}$ with classical constructions using basic combinatorial parameters: number of blocks, block size, replication, and pairwise concurrence. No statistical optimality criteria are used here.

6.3.1. Comparison Framework

We examine three representative depths:

• Shallow recursion ($n=1$): large blocks with limited diversity;
• Midpoint recursion ($n=n^{*}$): maximal number of blocks;
• Full recursion ($n=m$): a trivial design of singletons.

As reference constructions, we consider:

• Projective line design: the point-line incidence structure of $PG(7,2)$;
• Affine line design: the point-line incidence structure of $AG(7,2)$;
• Complementary recursive design: obtained by replacing n with $m-n$.

6.3.2. Numerical Comparison for $PG(7,2)$

The corresponding parameters are listed in

Table 3. Structural comparison between recursive designs from $PG(7,2)$ and related classical projective and affine line designs.

Design	Blocks	Block Size	Replication	Remarks
Recursive family
$n=1$	255	$2^6=64$	64	Large blocks, limited diversity
$n=4$	200,787	$2^3=8$	6299	Maximum number of blocks
$n=7$	255	1	1	Trivial design
Classical constructions
Projective lines ($PG(7,2)$)	10,795	3	127	Each pair of points lies on a unique line
Affine lines ($AG(7,2)$)	8128	2	127	Decomposes into parallel classes
Complementary ($n=3$)	97,155	16	6096	Dual to $n=5$

.

For the recursive rows, replication values are computed from the identity $bk=vr$, where $v=255$ is the number of points of $PG(7,2)$, assuming point-regularity of the construction.

The midpoint design ($n=4$) contains $200,787$ blocks, compared with $10,795$ for the projective line design. This reflects the concentration of subspaces near the middle dimension of the lattice.

As n increases, block size decreases exponentially ($k_n^{*}=2^{7-n}$), while the number of blocks follows a concave quadratic law. The maximum occurs at the midpoint.

The designs at depths n and $m-n$ have the same number of blocks and complementary block sizes, as a consequence of the symmetry of Gaussian binomial coefficients.

Projective and affine line designs are highly regular, with small block sizes and controlled pairwise structure. The recursive construction behaves differently, producing a much larger number of blocks by exploring a wider portion of the subspace lattice.

In the projective case, any two distinct points lie on exactly one line, while in the affine case a pair appears in at most one block. For the recursive construction, numerical checks indicate that pairwise concurrence remains bounded, consistent with the nested subspace structure.

Overall, the recursive family occupies a different regime: instead of emphasizing pairwise balance, it favors a large number of configurations arising from the symmetry of projective subspaces.

6.4. Practical Implications of the Trade-Off

The desirability function

$$D(p,m,n)={\displaystyle \frac{4((m+1)n-n^2)}{{(m+1)}^2}}$$

provides a simple way to select the recursion depth under basic constraints. We illustrate this with a few typical situations.

6.4.1. Scenario 1: Maximizing the Number of Blocks

If the objective is to maximize the number of admissible blocks, the optimal choice is the midpoint

$$n^{*}={\displaystyle \frac{m+1}{2}}.$$

For example, when $m=10$ and $p=2$, this corresponds to $n=5$ or $n=6$. The resulting number of blocks is on the order of $2^{34}$, as indicated in Table 2.

6.4.2. Scenario 2: Constraints on Block Size

Suppose that block size is bounded by $K_{\max }$. Since

$$k_n^{*}=p^{m-n},$$

this imposes the condition

$$n\ge m-{log}_p{K}_{\max }.$$

Within this range, one selects the value of n that maximizes $D(p,m,n)$.

For instance, with $m=10$, $p=2$, and $K_{\max }=32$, the constraint gives $n\ge 5$. The values $n=5$ and $n=6$ both yield high desirability (approximately $0.992$), with block sizes 32 and 16, respectively.

6.4.3. Scenario 3: Stability near the Optimum

The function $D(p,m,n)$ varies slowly near its maximum. A second-order expansion gives

$$D\left(n\right)\approx 1-{\displaystyle \frac{2}{{(m+1)}^2}}{\left(n-{\displaystyle \frac{m+1}{2}}\right)}^2.$$

This shows that values of n close to $n^{*}$ remain near-optimal.

For example, when $m=50$, depths within $\pm 5$ of the midpoint still yield desirability above $0.98$. This allows some flexibility when additional constraints are imposed. This behavior is illustrated in Figure 3, which shows both the scale-invariant structure and the flatness of the desirability function near the optimum.

6.5. Discussion and Synthesis

The numerical results are consistent with the theoretical framework developed in Section 2 and Section 3.

First, the exact enumeration confirms that the quadratic exponent $g\left(n\right)=nm-n^2+n$ provides an accurate description of the dominant growth. The residual terms remain bounded and do not affect the location of the maximum.

Second, the desirability function reflects the symmetry and concavity of the scaling law. The maximum occurs at the midpoint $n^{*}=(m+1)/2$, and for even m the two adjacent integer depths yield nearly identical values. This behavior is visible in both the tables and the plots.

Third, the structural comparison shows that the recursive family behaves differently from classical line-based designs. Projective and affine line designs have small block sizes and strong pairwise regularity. In contrast, the recursive construction produces a much larger number of blocks, especially near the midpoint depth. This reflects the distribution of subspaces in the projective lattice.

Finally, the practical scenarios illustrate how the desirability function can be used to select the recursion depth under simple constraints, such as bounds on block size. The flatness of the function near the maximum allows some flexibility without significant loss in the structural criterion.

7. Conclusions

We studied recursive constructions in finite projective geometries and showed that the recursion depth admits a natural optimal value determined by the proposed quadratic scaling law.

This function is strictly concave, and its maximum occurs at the midpoint $n^{*}=(m+1)/2$. This provides a simple criterion for selecting the recursion depth.

The analysis also shows that the growth of the number of blocks is governed by Gaussian binomial coefficients, and that the symmetry of these coefficients leads to the symmetry of the construction with respect to n and $m-n$.

The desirability function introduced in this work offers a convenient way to compare different depths using a normalized scale. It captures the same concavity and identifies the midpoint as the optimal region.

The comparison with projective and affine line designs indicates that the recursive family belongs to a different combinatorial regime. Classical designs emphasize small blocks and strong local regularity, whereas the recursive construction generates a much larger number of configurations near the midpoint.

Several directions remain open. The same analysis could be extended to other families of geometries, such as symplectic or unitary spaces. It would also be of interest to study the incidence structure of the recursive designs in more detail and to determine under which conditions they satisfy additional regularity properties.