Zig-Zag Encodings: Structures and Applications

• Zig-zag encodings are formal constructions that use alternating arrangements to represent algebraic, topological, combinatorial, or coding-theoretic structures.
• They employ methods such as recurrence relations, matrix representations, and constructible cosheaves to encode complex geometric and computational information.
• Applications span superconducting circuits, persistent homology, distributed storage codes, and dimer phase analysis, enhancing computational and design efficiency.

Searching arXiv for relevant papers on zig-zag encodings across domains. Zig-zag encodings are formal constructions in which a zig-zag geometry, ordering, filtration, or path family is used to represent algebraic, topological, combinatorial, or coding-theoretic structure. In recent arXiv literature, the expression appears in several technically distinct senses: encoding an orbital zig-zag topological model into superconducting circuits built from polarization transmons; encoding square-lattice zig-zag shapes by recurrence sequences and matrix recurrences; encoding zig-zag persistence by constructible cosheaves, entrance paths, and apex representatives; encoding parity and repair structure in zigzag array codes and shift-and-XOR fountain codes; and encoding dimer boundaries, spectral data, and phase structure by zig-zag paths in periodic bipartite graphs (Konopleva et al., 23 Oct 2025, Németh et al., 2021, Grady et al., 2021, Dey et al., 24 Feb 2025, Zhang et al., 27 Sep 2025, Nozaki, 2016, Berggren et al., 5 May 2026).

1. Terminological scope and common structural pattern

The literature uses “zig-zag encoding” for constructions built on alternating or staggered organization. The carrier object may be a 1D orbital chain with alternating couplings, a square-grid strip that zig-zags downward to the right, a zig-zag poset whose arrows alternate direction, a row-indexing group whose cosets determine repair reads, or a finite dimer domain whose boundary is formed by zig-zag paths. In this literature, the term does not denote a single standardized algorithm or invariant.

A common formal motif is the replacement of an ambient object by a zig-zag-indexed representation. In the combinatorial setting, shortest-path counts on a square $k$-zig-zag digraph are encoded by sequences $A_j^{(k)}$ and $Z_j^{(k)}$ satisfying explicit homogeneous recurrences. In persistence theory, a zig-zag filtration is encoded as a constructible cosheaf on stratified $\mathbb{R}$, and then as a representation of an entrance path category. In storage coding, a parity symbol is built by group-based permutations of row indices, while in fountain coding an encoded packet is built from shifted source packets through monomial labels in $z$. In the dimer setting, zig-zag paths carry the Newton polygon, spectral curve, and admissible finite-domain geometry (Németh et al., 2021, Grady et al., 2021, Zhang et al., 27 Sep 2025, Nozaki, 2016, Berggren et al., 5 May 2026).

This multiplicity matters because identical vocabulary masks different mathematical roles. In some papers the zig-zag object is the encoded structure itself; in others it is the index category, the boundary condition, the repair schedule, or the physical hardware layout.

2. Orbital zig-zag models and superconducting-circuit encodings

In the superconducting-topological setting, “zig-zag encodings” means taking an abstract orbital zig-zag topological model and encoding it into the physical degrees of freedom of a superconducting circuit built from polarization transmons. The zig-zag model is a 1D chain of sites, each hosting two degenerate dipole orbitals $p_x$ and $p_y$, with nearest-neighbor couplings alternating between strong $\sigma$-bonding $t_{\parallel}$ and weak $\pi$-bonding $A_j^{(k)}$0. At $A_j^{(k)}$1 and with nearest-neighbor couplings only, the model factorizes into two SSH chains with opposite staggering patterns, and the band energies are

$A_j^{(k)}$2

The proposed hardware realization uses polarization transmons hosting two degenerate dipole modes and one quadrupole mode, with the two dipoles having fundamental frequency $A_j^{(k)}$3, $A_j^{(k)}$4 (Konopleva et al., 23 Oct 2025).

The encoding is implemented by arranging polarization transmons in a zig-zag geometry so that the two degenerate dipole modes of each transmon map to the orbital labels $A_j^{(k)}$5 and $A_j^{(k)}$6. In second-quantized language, each site carries bosonic creation operators $A_j^{(k)}$7 and $A_j^{(k)}$8, and the physical electrode layout plus capacitive/inductive couplings reproduce alternating nearest-neighbor hopping and long-range cross-polarization next-nearest-neighbor couplings $A_j^{(k)}$9. The finite-chain Hamiltonian is written as

$Z_j^{(k)}$0

with $Z_j^{(k)}$1 (Konopleva et al., 23 Oct 2025).

The extended model remains chiral, with winding number

$Z_j^{(k)}$2

and supports topologically nontrivial phases with $Z_j^{(k)}$3, trivial phases with $Z_j^{(k)}$4, and transition lines where the bulk gap closes. The phase diagram is mapped numerically and analytically using inverse participation ratios and topological invariants; the work demonstrates in-gap localized trivial and Tamm edge states, and electromagnetic modeling of linearized meta-atoms shows that the proposed arrangement closely reproduces the extended zig-zag model (Konopleva et al., 23 Oct 2025).

3. Square zig-zag shapes, path counting, and sequence encodings

In combinatorics, zig-zag encoding is realized by assigning to vertices of a square $Z_j^{(k)}$5-zig-zag digraph the number of shortest paths from a base vertex, then reading integer sequences along columns and zig-zag paths. The underlying infinite strip has width $Z_j^{(k)}$6 and is produced by translating a bottom layer one unit right and one unit down at each step. Vertex values $Z_j^{(k)}$7 satisfy the local rule

$Z_j^{(k)}$8

For $Z_j^{(k)}$9, the partial-sum identity

$\mathbb{R}$0

shows that each new row is a cumulative summation of the previous row, truncated by the strip width (Németh et al., 2021).

This construction yields two principal encoded sequence families. The right-down diagonal sequences $\mathbb{R}$1 satisfy a common homogeneous recurrence of order $\mathbb{R}$2,

$\mathbb{R}$3

while the zig-zag sequences $\mathbb{R}$4, obtained by interleaving two adjacent columns, satisfy

$\mathbb{R}$5

For $\mathbb{R}$6, $\mathbb{R}$7 is the Fibonacci sequence. For $\mathbb{R}$8, the diagonal sequences satisfy $\mathbb{R}$9, and multiple examples coincide with OEIS entries such as A080937, A094790, A094789, and A005021 (Németh et al., 2021).

The entire row-to-row evolution is encoded by the matrix recurrence $z$0, where $z$1 and $z$2 is the partial-sum matrix. Its characteristic polynomial $z$3 obeys $z$4, and the resulting recurrences provide a compact algebraic encoding of the zig-zag geometry (Németh et al., 2021).

4. Zig-zag persistence, cosheaf encodings, and apex representatives

In topological data analysis, a zig-zag filtration is a sequence

$z$5

and a zig-zag persistence module is a functor $z$6 where $z$7 is a zig-zag poset whose Hasse diagram is homeomorphic to a closed interval, a half-open interval, or $z$8. One major encoding theorem identifies zig-zag modules with constructible cosheaves on stratified $z$9: the canonical zig-zag poset $p_x$0 is isomorphic to the combinatorial entrance path category $p_x$1, and

$p_x$2

The same framework supports augmented persistence modules that retain instantaneous events and leads to explicit $p_x$3-theory computations, including

$p_x$4

for 1-dimensional combinatorial manifolds (Grady et al., 2021).

A complementary algorithmic encoding converts an arbitrary zig-zag filtration into a levelset zigzag of a real-valued function on a prism complex, producing a Mayer–Vietoris pyramid of spaces and an infinite strip of homology groups. The indecomposable diamond summands of this strip have bottom-most origins called apexes. Apex representatives are obtained by lifting cycles computed from ordinary persistence on a cone filtration to cycles in the prism; these apex representatives map back to the levelset zigzag and then to the original zig-zag barcode representatives. The lifting algorithm for a $p_x$5-dimensional cycle takes $p_x$6 time, and zigzag representatives can then be recovered in $p_x$7, where $p_x$8 is the size of the output (Dey et al., 24 Feb 2025).

These two lines of work use “zig-zag encoding” in different but compatible ways: one is categorical and functorial, the other algorithmic and representative-level. Both treat the zig-zag not as incidental notation but as the primary indexing structure of persistence information.

5. Coding-theoretic encodings: array codes and fountain codes

In distributed storage, zig-zag encodings are embodied by zigzag array codes. An $p_x$9 MDS array code is stored as an $p_y$0 array over a field $p_y$1, with row indices in a finite additive group $p_y$2; the paper’s central case is $p_y$3, $p_y$4. For parity node $p_y$5, a tuple $p_y$6 defines row permutations, and the parity symbol in row $p_y$7 is a linear combination $p_y$8. Over characteristic-two fields, explicit Cauchy-type coefficients $p_y$9, with $\sigma$0 distinct and $\sigma$1, guarantee the MDS property and decouple the dependence among $\sigma$2, $\sigma$3, and $\sigma$4. The same work introduces skip cost and repair-fragmentation ratio, and constructs families with zero skip cost whose rates approach $\sigma$5, and families with bounded skip cost whose rates approach $\sigma$6 and $\sigma$7, while preserving optimal access and optimal rebuilding ratio (Zhang et al., 27 Sep 2025).

In fountain coding, zig-zag encoding means generating output packets by shifting some source packets and then combining them via bit-wise XOR. With source polynomials $\sigma$8, a zigzag decodable code has encoded packets

$\sigma$9

The proposed fountain architecture is a precode-plus-inner-code system: an LDPC precode first produces precoded packets, and then each output packet is formed as

$t_{\parallel}$0

where degree $t_{\parallel}$1 is drawn from $t_{\parallel}$2, shift amounts are drawn from $t_{\parallel}$3, normalized by subtracting the minimum, and the $t_{\parallel}$4 are stored in the header. The decoder uses packet-wise peeling and, if necessary, bit-wise peeling on a factor graph with monomial edge labels. The paper proves

$t_{\parallel}$5

so the decoding erasure probability is no worse than that of the corresponding Raptor code; simulations show lower decoding erasure rate, lower space decoding complexity, and improved overhead relative to Raptor coding systems (Nozaki, 2016).

The shared feature of these coding constructions is that zig-zag offsets are not merely combinatorial decoration. They determine linear constraints, repair locality, access contiguity, and decoding dynamics.

6. Zig-zag paths in dimer models, AZ graphs, and phase structure

In the dimer literature, zig-zag paths are the central encoding device. For a minimal $t_{\parallel}$6-periodic planar bipartite graph $t_{\parallel}$7, a zig-zag path is an oriented path that turns maximally right at white vertices and maximally left at black vertices. Each strand has a homology class in $t_{\parallel}$8, and these classes determine the sides of the Newton polygon $t_{\parallel}$9. For each side $\pi$0, the parallel class $\pi$1 consists of zig-zag paths with $\pi$2. This correspondence classifies minimal periodic graphs up to local moves preserving dimer statistical mechanics (Berggren et al., 5 May 2026).

Astroidal zig-zag graphs are finite subgraphs whose boundaries are formed by zig-zag paths, one per side of $\pi$3, in an order reversing the cyclic order of the polygon. Their admissibility is expressed through a boundary divisor $\pi$4 defined from the discrete Abel map, and the condition $\pi$5. For a Newton polygon with $\pi$6 sides, the paper constructs an $\pi$7-dimensional family of such finite subgraphs and obtains an explicit inverse Kasteleyn matrix for any Fock weighting. The inverse is given by a double contour integral on the corresponding spectral curve; for periodic weightings, the asymptotic analysis establishes phase separation into frozen, rough, and smooth regions, gives an explicit parametrization of the arctic curve, computes the deterministic limit of the height function, and proves convergence of local dimer correlations to the translation-invariant Gibbs measure of the slope predicted by the limit shape (Berggren et al., 5 May 2026).

This use of zig-zag encoding is geometrically distinct from the persistence and coding settings, but formally analogous. Boundary paths encode finite-domain geometry, the angle function on zig-zag strands encodes spectral data, and the real locus of the spectral curve parametrizes the arctic curve. In the special case where the Newton polygon is the unit square, the corresponding AZ graph is the Aztec diamond with its size as the parameter (Berggren et al., 5 May 2026).

Across these works, zig-zag encodings are therefore best understood as a family of domain-specific representation schemes unified by a common structural idea: alternating, staggered, or path-based zig-zag organization is used to transport information from one mathematical or physical description to another. The encoded content may be a topological Hamiltonian, a recurrence family, a persistence module, a parity architecture, or a dimer phase diagram, but in each case the zig-zag is the device that makes the representation explicit, computable, and structurally constrained.