Folding-Preserving Transpose

• Folding-preserving transpose is a transformation that reorders multidimensional arrays while preserving essential algebraic and combinatorial invariants defined by lattice tiling and directional walks.
• It relies on precise methods such as lattice tiling, directional walks, and gcd-based algebraic conditions to ensure every element is uniquely traversed under folding operations.
• Applications span error-correcting codes, network scheduling, and graph embeddings, where maintaining structure under transposition is critical for optimal performance and reliability.

The folding-preserving transpose is a structural and combinatorial concept arising in several mathematical and computational domains, notably multidimensional coding theory, algebraic representation theory, and geometric group theory. The term refers, in broad terms, to a transformation—such as coordinate or matrix transposition, network symmetry, or duality operation—that reorders or reflects objects (arrays, graphs, modules, cube complexes) while preserving essential combinatorial or algebraic properties established through a folding procedure. In the canonical context of multidimensional coding, it is defined through lattice tiling, directional walks, and strict algebraic criteria guaranteeing the maintenance of ordered structure under transposition. Analogous ideas appear in network communication theory as design-invariant scheduling, in algebra as module duality, and in cubical and median geometry as reconfigurations through folding/swellings that maintain key metric and subgroup representations.

1. Folding via Lattice Tiling and Directional Walks

The generalized folding operation in multidimensional coding is defined by a triple $(A, S, \delta)$ where $A$ is a lattice tiling (with generator matrix $G$) of a discrete $D$-dimensional shape $S$, and $\delta = (d_1, d_2, \dots, d_D)$ is a nonzero ternary direction vector ($d_i \in \{-1, 0, +1\}$). The folding constructs an ordered traversal ("folded-row") through $S$ by recursively stepping in the direction $\delta$ and applying a correction $c(\cdot)$ derived from the lattice. The process completes a cycle and visits every element of $S$ exactly once if \begin{align*} (|S| \cdot d_1, \dots, |S| \cdot d_D) - c(|S| \cdot d_1, \dots, |S| \cdot d_D) = (0, \dots, 0), \end{align*} and for $0 < i < |S|$, the position vectors $$(i \cdot d_1, \dots, i \cdot d_D) - c(i \cdot d_1, \dots, i \cdot d_D)$$ are distinct (0903.1724, 0911.1745). In two dimensions, explicit conditions are stated in terms of the generator matrix $G = [v_{11}\; v_{12}; v_{21}\; v_{22}]$, e.g., $\gcd(v_{22} - v_{21}, v_{11} - v_{12}) = 1$ for $\delta = (+1, +1)$.

2. Enumeration and Typology of Folding Operations

For a $D$-dimensional grid, the number of fundamentally distinct folding operations (up to reversal) is $(3^D - 1)/2$, due to the set of nonzero ternary direction vectors modulo taking the reverse direction (0903.1724). Each $\delta$ configures a unique ordering of the sequence's elements in the multidimensional space, which is critical in applications requiring specific local structural or error properties.

3. Necessary and Sufficient Algebraic Conditions

The existence of folding is governed by strict algebraic criteria, often involving greatest common divisor (gcd) conditions or solving lattice equations via Cramer’s rule: $$\sum_{j=1}^D a_j (v_{j1}, v_{j2}, ..., v_{jD}) = (|S|d_1, ..., |S|d_D)$$ with solution parameters $q_1, ..., q_p$, and $T$, then ensured by

$$\gcd(q_1, ..., q_p) = 1 \quad\text{and}\quad \gcd(T, |S|) = 1$$

(0911.1745). These criteria ensure that folded traversals do not "cycle too early"—i.e., prematurely revisit points—and that the lattice’s periodicity is correctly matched to the direction $\delta$. Maintenance of these conditions under coordinate reordering is the core requirement of the folding-preserving transpose: if they remain intact after a symmetry or transpose, the combinatorial structure is said to be preserved.

4. Folding-Preserving Transpose: Definition and Algebraic Characterization

A folding-preserving transpose is any transformation (coordinate transposition, isometry, matrix transpose, symmetry mapping, or schedule) that maps $(A, S, \delta)$ to $(A', S', T(\delta))$, where $T$ permutes coordinates or reindexes the underlying lattice/system, but the algebraic conditions for folding remain valid in the new configuration (0911.1745, 0903.1724). For rectangular arrays, this is the classical matrix transpose; for more general forms (non-rectangular arrays, complex tilings), it is a linear or symmetry operator for which the necessary gcd conditions and structural walk properties survive.

In symmetric network scheduling and communication theory, the analog is a schedule or configuration that leverages graph/processor symmetry (vertex transitive or Cayley digraphs), such that packet movement and resource usage templates (factorizations and generator paths) are invariant under network folding (symmetry operations). This ensures conflict-free transpose operations and schedule optimality (Faber, 2014).

5. Applications in Coding, Synchronization, and Graph Embeddings

Folded arrays constructed via these principles are essential in designing multidimensional error-correcting codes, pseudo-random arrays, and distinct-difference configurations for synchronization (0903.1724, 0911.1745). In error-correcting coding, a fold-preserving transpose allows for reorientation (dualization) of the code, maintaining burst-error detection or correction capabilities across array transformations. In synchronization and pseudo-random array construction, lattice folding (and its transposes) guarantees window properties and cyclic autocorrelation, properties vital for combinatorial designs and network keys.

In scalable graph representation, transpose proximity, defined via $s_u \cdot t_v \approx P(u, v) + P^T(v, u)$, leverages both standard and transposed graph random-walk proximities to preserve in- and out-degree distributions in embeddings. This bipartite matching (original plus transposed graph proximities) is essential to avoid conflicting optimization objectives and degree distortion—particularly in large, sparse, directed graphs—and underlies recent scalable algorithms such as STRAP (Yin et al., 2019).

6. Algebraic and Geometric Analogues: Dual Modules and Folding in Cubical Geometry

Classical module theory provides an algebraic analogue: any matrix $a \in M_n(F)$ is conjugate to its transpose $a^T$ by a symmetric matrix $g$, with $g a g^{-1} = a^T$ and $g^T = g$ (Madsen et al., 2019). For central simple algebras with involution, the generalization is $g a g^{-1} = 0(a)$ with $0(g) = \epsilon(0) \cdot g$. The folding-preserving feature here is the compatibility of structure under transpose (or involution) such that dual module representations and contragredient representations are maintained.

In geometry, folding and transposition are reshaped in the theory of CAT(0) cube complexes and median graphs (Ben-Zvi et al., 2020, Genevois et al., 2023). Stallings folding generalizes to cube identifications and attachments that algorithmically yield local isometries representing group substructures. Sageev–Roller duality further "transposes" the combinatorial content by swapping paths and hyperplanes, constructing the cubical convex hull as an isometric embedding—preserving the subgroup and local geometry data in the dual (transpose-like) structure. Median graph theory, extending cube complex folding, introduces foldings and swellings that canonically factor any parallel-preserving map into an isometric embedding, embodying the folding-preserving transpose on a combinatorial median space.

7. Structural and Practical Implications

The folding-preserving transpose is crucial in algorithmic construction, analytic invariance, and modular extension of structures across transformations:

• In multidimensional code design, it guarantees that combinatorial and error properties survive array orientation changes.
• In network scheduling, it yields optimality and uniformity across symmetric architectures.
• In graph embeddings, it preserves degree distributions and avoids conflicting proximities, enabling scalable, interpretable representations.
• In geometric group theory and median/cube complex geometry, it underpins subgroup algorithmics, local metric preservation, and structural convexity through canonical factorization by foldings and swellings.

This synthesis illustrates that folding-preserving transpose, in all its incarnations, is a structural guarantee: a transformation that retains critical combinatorial, algebraic, or metric invariants across reordering, symmetry, or dualization—empowering multidimensional construction, efficient communication schedules, robust algebraic dualities, and algorithmically tractable geometric representations.