Perfect Matroid Designs in Matroid Theory

• Perfect matroid designs are matroids in which every flat of a fixed rank exhibits the same cardinality, ensuring a uniform structure across the lattice of flats.
• They provide a unified framework that bridges classical block designs with projective, affine, and q-analogues, enabling precise incidence counting and design parameter derivation.
• Applications extend to cyclic flat compression, basis-relative enumeration, and connections with Chow rings and mixed Eulerian numbers for deeper combinatorial insights.

A perfect matroid design (PMD) is a matroid in which all flats of a given rank have the same cardinality. In the literature this common cardinality is variously denoted by $k_i$, $f_i$, $\alpha_i$, or $n_i$, depending on the source. The condition imposes a strong regularity on the lattice of flats: rank alone determines flat size, so full flags of flats have a prescribed cardinality profile. PMDs serve as a common framework for classical block designs, projective and affine $q$-analogues, and several later constructions in matroid theory, including cyclic-flat methods for Tutte polynomials, $q$-matroid analogues, and Chow-ring intersection invariants (Eberhardt, 2014, Zumbrägel, 2016, Byrne et al., 2020, Katz et al., 2023).

1. Definition, notation, and basic examples

In its classical form, a PMD is a matroid $M$ of rank $r$ such that for each $i\in\{0,\dots,r\}$ there exists a number $k_i$ with the property that every flat of rank $f_i$0 has cardinality $f_i$1. Equivalent formulations in the literature say that all $f_i$2-flats have the same size $f_i$3, or that along any full flag

$f_i$4

one has $f_i$5 for fixed integers $f_i$6. These are the same regularity condition expressed in different notational conventions (Kordecki, 2013, Zumbrägel, 2016, Katz et al., 2023).

The standard examples are equally consistent across the sources. Free matroids provide the classical set-theoretic model: every $f_i$7-flat is an $f_i$8-subset, so the type is $f_i$9. Vector matroids over $\alpha_i$0 provide the linear model, with $\alpha_i$1 when $\alpha_i$2-flats are vector subspaces. After geometrization, projective geometries $\alpha_i$3 have flat cardinalities

$\alpha_i$4

and affine geometries $\alpha_i$5 give another PMD family with $\alpha_i$6-flat cardinalities $\alpha_i$7. Other examples mentioned in the literature include uniform matroids, some Steiner systems, and Deza’s triffids (Kordecki, 2013, Zumbrägel, 2016).

This regularity is stronger than mere homogeneity of the ground set. It constrains both the sizes of flats and the incidence structure between rank levels. A plausible implication is that PMDs form a natural class in which global invariants can often be reduced to rank-wise parameters.

2. PMDs as ambient spaces for designs

A central use of PMDs is to treat designs as collections of flats inside a matroid. If $\alpha_i$8 is a PMD of rank $\alpha_i$9 and type $n_i$0, a $n_i$1-$n_i$2 design in $n_i$3 is a collection $n_i$4 of $n_i$5-flats such that each $n_i$6-flat is contained in exactly $n_i$7 members of $n_i$8. In this formulation, ordinary block designs arise from the free matroid, projective $n_i$9-analogues arise from projective geometry, and affine $q$0-analogues arise from affine geometry (Zumbrägel, 2016).

The PMD framework also yields the usual derived incidence parameters. If $q$1 is a PMD of rank $q$2 and type $q$3, then any $q$4-$q$5 design is also an $q$6-$q$7 design for each $q$8, with

$q$9

This is the matroidal analogue of the usual parameter relations for block designs, expressed entirely through flat cardinalities (Zumbrägel, 2016).

Affine geometry illustrates the flexibility of the framework. The affine space $q$0 has an associated affine matroid of rank $q$1, whose $q$2-flats are $q$3-dimensional affine subspaces. A $q$4-$q$5 affine design is therefore a collection of affine $q$6-flats uniformly covering affine $q$7-flats. The relation between projective and affine designs is especially tight: if $q$8 is a $q$9-$M$0 subspace design in an $M$1-dimensional $M$2-vector space $M$3, then translating all blocks by the translation group $M$4 produces a $M$5-$M$6 affine design in $M$7; conversely, blocks containing $M$8 in such an affine design recover a $M$9-$r$0 subspace design (Zumbrägel, 2016).

The same paper records existence results for affine Steiner systems, including affine $r$1 systems and an affine $r$2 in $r$3. It also discusses a Singer-cycle invariant affine $r$4 with 273 parallel classes and its coding-theoretic use. This suggests that PMDs are not only an organizing language for design theory but also a useful ambient structure for affine and projective coding constructions (Zumbrägel, 2016).

3. Cyclic flats, condensed configurations, and Tutte polynomial rigidity

A major structural result for PMDs arises from the lattice of cyclic flats. For a matroid $r$5, a cyclic flat is a flat $r$6 such that the restriction $r$7 has no coloops; equivalently, every element of $r$8 lies in a circuit of $r$9. The set $i\in\{0,\dots,r\}$0 of cyclic flats forms a lattice under inclusion. The paper “Computing the Tutte Polynomial of a Matroid from its Lattice of Cyclic Flats” develops cloud and flock polynomials on $i\in\{0,\dots,r\}$1 and proves that the rank generating polynomial $i\in\{0,\dots,r\}$2, hence also the Tutte polynomial $i\in\{0,\dots,r\}$3, is determined by the configuration of $i\in\{0,\dots,r\}$4: the abstract lattice $i\in\{0,\dots,r\}$5 together with the rank and cardinality of each cyclic flat (Eberhardt, 2014).

The same work introduces a further compression, the condensed configuration. A condensation is a partition $i\in\{0,\dots,r\}$6 of $i\in\{0,\dots,r\}$7 such that rank and cardinality are constant on blocks and the inclusion numbers

$i\in\{0,\dots,r\}$8

are independent of the chosen $i\in\{0,\dots,r\}$9. Theorem 5.3 of that paper shows that $k_i$0 can be computed from any condensed configuration. For PMDs, this compression is especially effective (Eberhardt, 2014).

If $k_i$1 denotes the set of rank-$k_i$2 flats of a PMD and $k_i$3 their common cardinality, then a rank-$k_i$4 flat is cyclic if and only if $k_i$5. For $k_i$6 and $k_i$7, the number of rank-$k_i$8 flats contained in $k_i$9 is

$f_i$00

Consequently, the collection

$f_i$01

is a condensation of $f_i$02, and the corresponding condensed configuration is determined entirely by the sequence $f_i$03 (Eberhardt, 2014).

This yields a reproof of Mphako’s theorem: for a perfect matroid design, the rank generating polynomial $f_i$04, and therefore the Tutte polynomial, depends only on the cardinalities and ranks of its flats. The result is stronger in context, because it is subsumed by a general cyclic-flat formalism valid for arbitrary matroids. The same perspective also clarifies why Shoda’s superexponential families of matroids with identical Tutte polynomial share the same configuration by construction (Eberhardt, 2014).

4. Symmetric PMDs and basis-relative enumeration

A more specialized class is the symmetric perfect matroid design (SPMD). In the formulation used for hyperplane and circuit enumeration, an SPMD is a rank-$f_i$05 matroid in which all flats of the same rank $f_i$06 are isomorphic as matroids, and the number of rank-$f_i$07 flats containing a fixed rank-$f_i$08 flat depends only on $f_i$09, not on the chosen flat. Projective and affine finite geometries are the prototypical examples (Kordecki, 2013).

This symmetry permits explicit basis-relative counting. Let $f_i$10 be a basis of an SPMD of rank $f_i$11. The number of hyperplanes $f_i$12 with $f_i$13 is

$f_i$14

where $f_i$15 is the rank-containment function. The number of elements $f_i$16 such that $f_i$17 is a circuit is

$f_i$18

where $f_i$19 is the number of elements in a rank-$f_i$20 flat (Kordecki, 2013).

In projective geometry $f_i$21, both counts simplify to

$f_i$22

In affine geometry $f_i$23, the number of hyperplanes avoiding a basis is

$f_i$24

while the number of points $f_i$25 such that $f_i$26 is a circuit is

$f_i$27

The projective equality reflects the point–hyperplane duality of projective space; the affine correction terms record the loss of full projective symmetry after deleting the hyperplane at infinity (Kordecki, 2013).

The same paper gives constructive algorithms for enumerating all such hyperplanes and all such basis-extending circuits in $f_i$28 and $f_i$29. In the projective case the algorithms proceed by selecting interior points on lines joining successive basis elements, or dually by intersecting suitable hyperplanes through coordinate intersections. In the affine case the constructions are obtained by embedding in projective space and removing the hyperplane at infinity (Kordecki, 2013).

5. $f_i$30-Perfect matroid designs

A different generalization replaces ordinary matroids by $f_i$31-matroids, where the underlying lattice is the lattice of subspaces of $f_i$32. A $f_i$33-matroid is a pair $f_i$34 with $f_i$35 defined on subspaces and satisfying the rank axioms

$f_i$36

The paper “Constructions of new matroids and designs over GF(q)” establishes a flat cryptomorphism for $f_i$37-matroids and defines a $f_i$38-perfect matroid design ($f_i$39-PMD) as a $f_i$40-matroid in which any two flats of the same rank have the same dimension (Byrne et al., 2020).

Its main source of examples is $f_i$41-Steiner systems. If $f_i$42 is an $f_i$43 system, then the family of all block intersections

$f_i$44

consists exactly of $f_i$45, the blocks $f_i$46, and all subspaces of dimension at most $f_i$47. This family satisfies the flat axioms and defines a $f_i$48-matroid whose rank function is explicitly computed; the induced $f_i$49-matroid is a $f_i$50-PMD (Byrne et al., 2020).

The induced $f_i$51-PMD supports new subspace-design constructions. If $f_i$52 denotes the independent subspaces of dimension $f_i$53, then

$f_i$54

is a $f_i$55-$f_i$56 design with

$f_i$57

If $f_i$58 denotes the circuits of dimension $f_i$59, then

$f_i$60

is a $f_i$61-$f_i$62 design with

$f_i$63

There is also a construction from circuits of dimension $f_i$64, with an explicit formula for the corresponding $f_i$65 (Byrne et al., 2020).

The principal application uses the only known $f_i$66-Steiner system with $f_i$67, namely $f_i$68. Applying the $f_i$69-PMD machinery yields, among other derived designs, a new subspace design with parameters

$f_i$70

This places $f_i$71-PMDs in direct continuity with the earlier classical role of PMDs as design-producing structures (Byrne et al., 2020).

6. Chow rings and perfect matroidal mixed Eulerian numbers

Recent work connects PMDs to intersection theory in the matroid Chow ring. For a loopless matroid $f_i$72 of rank $f_i$73, the Chow ring $f_i$74 carries divisor classes $f_i$75, and for nonnegative integers $f_i$76 with $f_i$77, the matroidal mixed Eulerian number is

$f_i$78

These numbers are valuative and satisfy a log-concavity relation of Khovanskii–Teissier type. They also recover the coefficients of the reduced characteristic polynomial via

$f_i$79

and they are related to specializations of the Tutte polynomial $f_i$80 (Katz et al., 2023).

For a PMD with flat sizes $f_i$81, the relevant specialization is

$f_i$82

The structural theorem here is a quadratic relation among consecutive hypersimplex classes: $f_i$83 It produces a recursion for perfect matroidal mixed Eulerian numbers and, in projective geometry, specializes to the recurrence for Nadeau–Tewari’s remixed Eulerian numbers (Katz et al., 2023).

A particularly simple closed form occurs for lopsided multi-indices, meaning those with

$f_i$84

If

$f_i$85

where $f_i$86 is the number of rank-$f_i$87 flats in a given rank-$f_i$88 flat, then

$f_i$89

for every lopsided $f_i$90. In the special case $f_i$91, the resulting numbers coincide, up to a factor $f_i$92, with the remixed Eulerian numbers of Nadeau and Tewari. This places PMDs inside a broader intersection-theoretic and Eulerian framework (Katz et al., 2023).

7. Terminological distinctions and adjacent frameworks

The classical expression “perfect matroid design” should be distinguished from other uses of the word “perfect” in nearby matroidal literatures. In locally repairable coding theory, an $f_i$93-matroid is called perfect when it attains the generalized Singleton bound

$f_i$94

That usage concerns optimal locality and distance, not equicardinality of flats, even though the resulting structure theorems involve cyclic flats arranged in a block-design-like pattern (Westerbäck et al., 2014).

A second distinct usage appears in the theory of linear spaces over perfect idylls. There, an idyll $f_i$95 is perfect when for every $f_i$96-vector set $f_i$97,

$f_i$98

Under that hypothesis, the paper constructs $f_i$99-linear spaces whose finite dependence sets are $\alpha_i$00-vector sets, so linear independence satisfies matroid independence axioms. The term “perfect” in this setting refers to exact orthogonality behavior of vector and covector sets, not to the flat-cardinality condition defining PMDs (Liu, 25 Jun 2026).

These terminological collisions matter because PMDs interact with many of the same objects—cyclic flats, vector-set realizations, design structures, and coding-theoretic constraints—without being reducible to any one of them. The classical PMD remains the design-theoretic notion: a matroid whose flats of the same rank all have the same size. Its significance lies in the fact that this seemingly simple equicardinality condition supports a wide range of exact formulas, from inclusion counts and design parameters to Tutte polynomial determination, $\alpha_i$01-analogues, and Chow-ring recursions (Eberhardt, 2014, Zumbrägel, 2016, Katz et al., 2023).