Cylinder Codewords: Structure & Applications

• Cylinder Codewords are structured objects in finite geometry, coding theory, and translation surfaces defined by unions of parallel affine subspaces with p-divisible properties.
• They underpin the Cylinder Conjecture by characterizing minimal support vectors as unions of parallel lines or higher-dimensional subspaces in projective settings.
• Their analysis employs structural decompositions and hyperplane-section lemmas to bridge discrete geometry with dynamics and combinatorial invariants.

A cylinder codeword is a structural object arising at the intersection of finite geometry, coding theory, and the study of translation surfaces. It encodes the highly regular combinatorial patterns found in configurations that generalize unions of parallel lines, “cylinders,” in finite vector or projective spaces. Such codewords appear as minimal support (smallest-weight) vectors in classes of divisible codes, as exact solutions to extremal set problems, and as the combinatorial invariants of translation surface cylinder decompositions. Their analysis rests on deep structural theorems and conjectures—foremost among these are the Cylinder Conjecture and its generalizations, which characterize the geometry and support of divisible objects in both classical and modern settings.

1. Cylinder Codewords in Finite Geometry and Coding Theory

A cylinder codeword, in the context of a finite geometry over $\mathbb{F}_p^n$, is any codeword in a $p$-divisible linear code whose support is a cylinder; that is, a union of $p$ parallel lines in the affine space $\mathbb{F}_p^3$ or, more generally, the union of $q$ parallel affine $(r+1)$-dimensional subspaces in $\mathbb{F}_q^v$. In these settings, a set $S \subseteq \mathbb{F}_p^3$ is called $p$-divisible if every affine hyperplane $H$ has $|S \cap H| \equiv 0 \pmod p$. A cylinder in $\mathbb{F}_p^3$ is specifically a union of $p$ pairwise disjoint parallel lines; its characteristic function is thus a $0$–$1$-valued function on the space, attaining value $1$ exactly on the points of the cylinder (Kiss et al., 14 Jan 2026).

In coding-theoretic terms, cylinder codewords span the space of all $p$-divisible codewords—this is because every $p$-divisible set in $\mathbb{F}_p^3$, and more generally every $q^r$-divisible set in higher dimensions, can be expressed as an $\mathbb{F}_p$-linear combination (and in certain cases $\mathbb{Z}$-linear combination) of characteristic functions of cylinders (Kiss et al., 14 Jan 2026, Kurz et al., 2020).

2. The Cylinder Conjecture and its Generalizations

The foundational structural question for cylinder codewords is encapsulated in the Cylinder Conjecture. The Strong Cylinder Conjecture, due to S. Ball (2008), asserts: If $S \subseteq \mathbb{F}_p^3$ is a $p$-divisible set of cardinality $p^2$, then $S$ must be a cylinder. In the language of codes, the conjecture claims: Any projective $[p^2, 4]_p$ code that is $p$-divisible has codeword supports equal to the union of $p$ parallel affine lines—the canonical cylinder codewords (Kiss et al., 14 Jan 2026, Kurz et al., 2020).

A generalization replaces $p$ by $q$, considers $q^r$-divisible sets, and extends to higher-dimensional projective space. The Generalized Cylinder Conjecture states that a spanning $q^r$-divisible set $S \subset \mathrm{PG}(v-1, q)$ with $|S|=q^{r+1}$ must be a (spanning) $(r+1)$-cylinder. The validity of this conjecture depends on the difference $v-r$, and can be reduced to the so-called "base case" $(v', 1, q)$ by a series of structural reductions and hyperplane-section lemmas (Kurz et al., 2020).

Counterexamples have been constructed in non-prime fields and special dimensions via subfield or field-reduction arguments. For prime $q$, the conjecture holds in all tested cases, now including $q \le 7$ (Kurz et al., 2020).

3. Structural Decomposition Theorems

Structural results demonstrate that every $p$-divisible multiset in $\mathbb{F}_p^3$ (of any total weight) can be decomposed as an $\mathbb{F}_p$-linear combination of cylinder characteristic functions (Kiss et al., 14 Jan 2026): $$f_S(x) = \sum_{i=1}^r a_i 1_{C_i}(x), \quad a_i \in \mathbb{F}_p$$ When the multiset size is $p^2$, this further refines to an integral decomposition: $$1_S(x) = 1_H(x) + \sum_{k=1}^m c_k \left(1_{L_k}(x) - 1_{L_k'}(x)\right), \quad c_k \in \mathbb{Z}$$ where $H$ is an affine plane and each $(L_k, L_k')$ is a pair of parallel lines. This efficiently characterizes $p$-divisible sets as constructed from cylinders and parallel-line differences (Kiss et al., 14 Jan 2026).

In projective geometry, the dual code $\mathcal{C}_1(n,q)^\perp$ (kernel of the line vs. point incidence matrix) admits cylinder codewords arising from the extension of small-weight codewords in $\mathcal{C}_1(2,q)^\perp$ to higher dimensions by a "cylinderization" procedure—explicitly, by projecting with respect to a disjoint subspace (the "vertex") (Adriaensen, 18 Jan 2026).

Table 1: Decomposition Paradigms for Cylinder Codewords

Setting	Decomposition	Citation
$\mathbb{F}_p^3$, $p$-divisible	$\mathbb{F}_p$-span of cylinders, $\mathbb{Z}$-comb. with planes/line-pairs	(Kiss et al., 14 Jan 2026)
$\mathrm{PG}(n,q)$, dual code	Cylinder codewords span minimum weight (under conjecture)	(Adriaensen, 18 Jan 2026)
$q^r$-divisible codes	Generalized cylinder codewords, spectral/LP bounds	(Kurz et al., 2020)

4. Cylinder Codewords and Minimal Weight in Dual Projective Codes

In the dual projective geometric code $\mathcal{C}_1(n,q)^\perp$, codewords of minimal weight are precisely cylinder codewords under certain conditions. For prime $q$, and for even $q$, all minimal-weight codewords can be constructed as cylinders with base a minimum-weight codeword in the plane ($\mathcal{C}_1(2,q)^\perp$) and extended along a complementary subspace (Adriaensen, 18 Jan 2026).

Explicitly, given a base codeword $c_0 \in \mathcal{C}_1(2,q)^\perp$ and vertex $\rho \cong \mathrm{PG}(n-3,q)$, the induced cylinder codeword $\mathrm{Cyl}_\rho(c_0)$ in $\mathcal{C}_1(n,q)^\perp$ has support equal to the union of $(n-2)$-flats through each point in the support of $c_0$ (excluding the vertex). The weight of such a codeword is $q^{n-2}$ times the weight in the plane.

For $q$ even, the connection with "even sets" (subsets meeting every line in an even number of points) further demonstrates that cylinders over hyperovals are the unique minimum-size even sets (Adriaensen, 18 Jan 2026).

5. Algorithms and Cylinder Codewords in Translation Surfaces

In the theory of square-tiled (translation) surfaces, cylinder codewords arise as permutation elements encoding the combinatorics of cylinder decompositions in rational directions (Tan, 2012). Each cylinder decomposition in slope $r$ is recorded as a "left code," a permutation in the symmetric group $S_n$ derived from the gluing data of the surface. These cylinder codewords enumerate all possible combinatorial types of directional cylinders and provide a dictionary between $SL_2(\mathbb{Z})$ orbits of the surface and conjugacy classes in $S_n \times S_n$.

The closed system of left codes is constructed using Farey addition and an explicit algorithm leveraging the additivity of codes under composition. These permutation-valued cylinder codewords encode topological and dynamical invariants of the surface and its moduli orbit (Tan, 2012).

6. Classification Results, Counterexamples, and Open Problems

Combinatorial, linear programming, and computer-aided methods have established cases of the Generalized Cylinder Conjecture for many parameters: it holds for all $q = 2,3,5,7$ in all dimensions, with precise classification available in these cases (Kurz et al., 2020). For non-prime fields $q = p^h$, field-reduction techniques produce "Baer-type" counterexamples not of pure cylindrical type, with precise bounds on the dimensions for which such counterexamples may appear.

For dual projective geometric codes, the reduction of the minimum-weight problem for $\mathcal{C}_1(n,q)^\perp$ entirely to the plane case ($n=2$) has simplified the classification of minimal supports, enabling a complete solution for even $q$ and providing a pathway for future classification in odd $q$ (Adriaensen, 18 Jan 2026).

Open problems include the complete characterization of minimal-weight codewords in dual projective codes over odd non-prime fields and full identification of all counterexamples to the Generalized Cylinder Conjecture, especially for $q$ non-prime and higher $r$.

7. Significance in Finite Geometry, Coding, and Dynamics

Cylinder codewords serve as a unifying concept across several disciplines:

• In finite geometry and extremal combinatorics, they identify maximal structures avoided (or forced) by divisibility conditions.
• In coding theory, their existence, classification, and decomposition properties enable precise structure theorems for divisible codes, with the support of any $q^r$-divisible codeword strongly constrained by the geometry of cylinders (Kiss et al., 14 Jan 2026, Kurz et al., 2020).
• In the dynamics of translation surfaces, cylinder codewords encode the entirety of combinatorial orbit data for square-tiled surfaces under $SL_2(\mathbb{Z})$ action (Tan, 2012).

The development and analysis of cylinder codewords thus continue to influence deep structural questions in mathematics, tying together discrete geometry, algebraic coding theory, and dynamical systems.