Position-Estimation Codes: Methods & Applications

• Position-estimation codes are structured constructs that embed positional metadata, allowing unique coordinate recovery or error localization from partial or corrupted data.
• They employ both combinatorial and binary (including wavelet-based) designs to manage grid delimitation and balance constraints, ensuring efficient decoding.
• These codes integrate local uniqueness, parity identification, and transformation techniques to reduce redundancy while improving error-correction in deletion and sensing applications.

Searching arXiv for relevant papers on position-estimation codes, position coding, and related estimation-code design.

Position-estimation codes denote code constructions in which a constrained observation is engineered to identify a location, a short location interval, or another position-related latent variable. In one classical discrete-mathematical sense, a position coding pattern is “an array of symbols in which subarrays of a certain fixed size appear at most once,” so a local window can be decoded into coordinates (0706.0869). In deletion-correcting coding theory, a position-estimation component approximates error positions before a local correction stage reconstructs the deleted symbols (Ye et al., 7 Jul 2025). A broader engineering usage also appears in sensing and wireless estimation, where designed tag responses or spatial pilot patterns function as codes that improve orientation or position recovery (Rammal et al., 2021).

1. Scope and formal interpretations

The literature uses closely related terminology in several technically distinct ways. The common thread is not a single channel model or decoder architecture, but the deliberate design of observable structure so that positional information can be inferred from partial data. In discrete patterns, the observation is a local subarray. In deletion correction, the observation is a corrupted word from which a short corruption interval must be inferred. In sensing-oriented design, the observation is a measured response induced by a deliberately chosen probing structure such as a tag response or a spatial pilot pattern (0706.0869).

A concise way to separate the main traditions is to distinguish the coordinate-decoding, error-localization, and probing-design viewpoints. The first decodes physical coordinates from local combinatorial uniqueness; the second localizes corruption in symbol sequences; the third designs measurement structure so that geometric state estimation is more accurate. The end-to-end mmWave positioning literature makes this third usage explicit by stating that the transmitter precoder is the “code” that shapes the spatial probing signal, while the receiver network is the estimator (Rivetti et al., 2022).

Family	Observation	Decoded quantity
Position coding pattern	Fixed-size local subarray	Coordinate
Error-position-estimation code	Sequence after deletions	Short interval containing corruption
Estimation-oriented probing code	Tag or pilot response	Position or orientation estimate

This taxonomy also prevents a common ambiguity. “Position” may refer either to a physical location in a 2D or 3D coordinate system or to the position of an error event in a symbolic sequence. Both usages are well established in the cited literature (Ye et al., 7 Jul 2025).

2. Local uniqueness and the combinatorics of coordinate recovery

The foundational definition in the combinatorial literature is the position coding pattern: an array in which every subarray of a prescribed size appears at most once. If a fixed-size window is unique, then observing that window identifies where it came from. The stronger exact-uniqueness analogue is the De Bruijn torus: an $(R,S;m,n)_k$-De Bruijn torus is a two-dimensional toroidal array with $R$ rows and $S$ columns in which every $m\times n$ array over a $k$-symbol alphabet appears exactly once (0706.0869).

The decoding philosophy is correspondingly uniform. A coordinate is encoded into a structured local pattern; a reader acquires a small subarray; the subarray is matched to a canonical pattern class; and the coordinate is reconstructed by lookup or algebraic inversion. In the non-wavelet construction, the notation

$$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$

is used to extract $2\times 2$ subarrays and decode them by parity and lookup (0706.0869).

The classical industrial exemplar discussed in this literature is the Anoto pattern used in the Fly Pentop system. Its dots lie on a raster, but each dot is slightly offset in one of four possible directions, and the pattern has the property that any $6\times 6$ array of dots appears only once. Each dot offset contributes one bit to the $x$-coordinate and one bit to the $y$-coordinate. The construction is organized around a cyclic binary main number sequence of length $R$0, which is a quasi De Bruijn sequence of order $R$1 (0706.0869).

A key intermediate object is the primary difference sequence. If adjacent columns are shifted by $R$2 and $R$3, the difference is

$R$4

The allowed values are then restricted to $R$5, and an explicit bijection

$R$6

is defined by

$R$7

These secondary sequences yield a multi-level digit decomposition of the coordinate. The paper further notes that

$R$8

which quantifies the repetition period of the combined secondary structure (0706.0869).

3. Binary constructions: wavelet and non-wavelet designs

Two new binary constructions in the same line of work make the design principles more explicit. The first is a binary wavelet-based position coding pattern using the matrix

$R$9

For a $S$0 matrix $S$1, the transform is $S$2. Coordinates are encoded into a $S$3 block in which the top two rows encode the $S$4-coordinate and the bottom two rows encode the $S$5-coordinate; the stored page pattern is obtained by applying the inverse transform. This code represents positions from $S$6 through $S$7, hence $S$8 positions, and appears as a $S$9 binary matrix. Its main stated limitation is that it requires gridlines or some form of delimiters to distinguish individual matrices (0706.0869).

The second construction is a delimiter-free non-wavelet binary code. For even $m\times n$0 and $m\times n$1, the paper states that one can create a binary position coding pattern in which $m\times n$2 sections are unique; the explicit construction is a $m\times n$3 binary array in which every $m\times n$4 section appears at most once. It is built from a fixed

$m\times n$5

a sequence $m\times n$6 of $m\times n$7 distinct $m\times n$8 arrays, and a derived sequence $m\times n$9 formed by combining the second column of one block with the first column of the next (0706.0869).

Decoding proceeds by isolating parity information and then applying lookup tables. A distinguished set

$k$0

identifies the within-block offset, because exactly one $k$1 subblock of a $k$2 window lies in $k$3. The remaining $k$4 blocks are decoded by membership in $k$5, $k$6, or transformed versions of them. The coordinate decomposition is explicit: the $k$7-coordinate is represented by two base-$k$8 digits plus a residue class modulo $k$9, and the $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$0-coordinate is represented analogously (0706.0869).

These binary constructions clarify an important point about position-estimation codes in the combinatorial sense: local uniqueness alone is not the entire design problem. Self-delimitation, parity localization, and low-complexity decoding tables are equally central. The wavelet design has clean algebraic structure but needs delimiters, whereas the non-wavelet construction integrates localization of the reading frame into the code itself.

4. Error-position-estimation codes for deletion channels

A second major meaning of position-estimation code arises in deletion correction. The coding-theoretic problem is not to recover physical coordinates but to localize where deletions occurred. The paper on bursty and localized deletions states that existing constructions employ a position-estimation component to approximate error positions, augmented by additional constraints that enable error-correction given the information about error positions. The new contribution is a position-estimation mechanism based on differential sequences whose differential sequences are strong-$$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$1-locally-balanced (Ye et al., 7 Jul 2025).

Two adversarial models are considered. A $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$2-burst-deletion deletes a contiguous substring of length at most $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$3. A $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$4-localized deletion deletes at most $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$5 symbols inside an unknown window of length $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$6. In both models, the critical first step is to determine a short interval containing the corruption; once the location is narrowed to a window of length $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$7, a separate local code can recover the deleted symbols (Ye et al., 7 Jul 2025).

The central algebraic device is the differential sequence transform. For $$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$8,

$$Z(i,j;m,n)= \left[ \begin{array}{cc} Z_{i,j} & Z_{i,j+n}\ Z_{i+m,j} & Z_{i+m,j+n} \end{array} \right]$$9

with

$2\times 2$0

when $2\times 2$1. This map is invertible and is a bijection onto

$2\times 2$2

A single deletion in $2\times 2$3 becomes a local replacement in $2\times 2$4: two adjacent symbols $2\times 2$5 are replaced by their mod-$2\times 2$6 sum $2\times 2$7. That locality is the key simplification (Ye et al., 7 Jul 2025).

The balance condition is then imposed in the differential domain. A sequence is strong-$2\times 2$8-locally-balanced if every substring of length $2\times 2$9 has $6\times 6$0-weight in

$6\times 6$1

This prevents long substrings from being too light or too heavy. In the decoding analysis, such bounds force any candidate index $6\times 6$2 consistent with the VT-derived equations to lie close to the true deletion location; otherwise the implied interval weight contradicts local balance (Ye et al., 7 Jul 2025).

Two further constraints turn localization into a code. The first is a VT-type constraint

$6\times 6$3

The second is an $6\times 6$4-weight congruence constraint

$6\times 6$5

The final burst-deletion code is

$6\times 6$6

where $6\times 6$7. The decoder uses the VT and sum constraints to derive $6\times 6$8 and $6\times 6$9, scans backward for a consistent index $x$0, proves that $x$1 lies within $x$2 of the true location, and then reduces the problem to $x$3-bounded local burst-error correction (Ye et al., 7 Jul 2025).

The redundancy improvement is explicit. For fixed $x$4 and $x$5, when $x$6 and $x$7, or $x$8 is even and $x$9, the paper constructs both $y$0-ary $y$1-burst-deletion correcting codes and $y$2-localized-deletion correcting codes with redundancy

$y$3

The abstract states that earlier best-known constructions achieved $y$4 for general $y$5, $y$6 for even $y$7 in the burst-deletion case, and $y$8 or $y$9 for localized deletions. The new mechanism is therefore both simpler and lower-redundancy in the stated regimes (Ye et al., 7 Jul 2025).

The same work also gives an efficient encoder that maps an arbitrary input sequence into one whose differential sequence is strong-$R$00-locally-balanced. The authors state that, to their knowledge, no prior algorithm for this specific task had been reported (Ye et al., 7 Jul 2025).

5. Estimation-oriented code design in sensing and communications

A broader engineering usage treats a code as a deliberately designed probing structure rather than a symbolic redundancy constraint. The backscatter-orientation paper “Coded Estimation: Design of Backscatter Array Codes for 3D Orientation Estimation” considers configurable backscatter tags, designs tag response codes to improve the accuracy of orientation estimation, proposes two code design criteria to minimize the difference between the true and estimated orientation, derives a lower bound on the worst-case error using Le Cam’s method, and reports simulations for line-of-sight only and multipath scenarios (Rammal et al., 2021).

A related development appears in end-to-end learned mmWave positioning. “Spatial Signal Design for Positioning via End-to-End Learning” can be viewed as learning a positioning codebook or spatial pilot design: the transmitter precoder is the “code” that shapes the spatial probing signal, and the receiver network maps the measured pilot response to angle of departure or position. In the reported experiments, the learned design matches CRB-optimized model-based beamforming under ideal conditions and is more robust under array spacing perturbations and mutual coupling (Rivetti et al., 2022).

These engineering examples differ from combinatorial position coding and deletion-position estimation in what is constrained. The object being designed is not a local symbolic pattern with an at-most-once property, nor a redundancy class in a $R$01-ary sequence, but a measurement-inducing structure. Even so, the design logic is analogous: shape the observable so that the inverse problem is better conditioned. That analogy is explicit in the wireless paper’s statement that the precoder is the code and the receiver network is the estimator (Rivetti et al., 2022).

The contrast with direct position estimation is also instructive. In OFDM-based cellular localization, direct position estimation searches over the position in the waveform domain rather than first compressing the signal into intermediate delay or Doppler estimates, and the paper reports that this method consistently outperforms the two-step approach under all evaluated conditions (Li et al., 4 Aug 2025). In cooperative MIMO-OTFS ISAC, a direct maximum-likelihood framework estimates target position in a common reference system rather than relying on local range and angle estimates at each base station (Pucci et al., 2024). These methods are not position-estimation codes in the narrow coding-theoretic sense, but they define the neighboring methodological landscape against which coded probing and coded observability should be understood.

6. Conceptual distinctions, misconceptions, and persistent design themes

A common misconception is that position-estimation codes always estimate absolute Cartesian coordinates. The literature considered here does not support that restriction. In position coding patterns, the decoded object is a coordinate in a larger array (0706.0869). In deletion-correcting codes, the decoded object is a short interval containing the corruption, after which a local code performs exact correction (Ye et al., 7 Jul 2025). In sensing-oriented code design, the decoded quantity may be orientation, angle of departure, or position, depending on the probing structure (Rammal et al., 2021).

A second misconception is that uniqueness alone suffices. The wavelet-based binary construction shows otherwise: it encodes $R$02 positions but requires gridlines or delimiters, so local uniqueness is not by itself enough for practical self-localization (0706.0869). Conversely, the non-wavelet construction integrates parity-class identification into the code, and deletion-position estimation integrates VT-type and $R$03-weight constraints with local balance and auxiliary bounded burst-error correction. In both domains, decodability depends on side structure in addition to the primary coding rule (0706.0869).

A third persistent theme is the movement from direct localization of the target quantity to localization via a transformed or designed representation. The differential map $R$04 in deletion correction is used because deletions become local replacements of adjacent symbols by their sum, making weighted-sum changes analyzable (Ye et al., 7 Jul 2025). The Anoto construction uses column shifts, a primary difference sequence, and secondary sequences rather than direct coordinate labels (0706.0869). End-to-end wireless positioning learns a probing codebook rather than analytically specifying a closed-form beam code (Rivetti et al., 2022). This suggests that position-estimation codes are often most effective when the observable domain is chosen so that locality, balance, or geometric sensitivity becomes easier to exploit.

Across these traditions, three technical invariants recur. First, the code must induce a decoding statistic whose ambiguity set is small: a unique local window, a short candidate corruption interval, or a well-conditioned measurement response. Second, the decoder typically combines a coarse localization stage with a refinement or correction stage. Third, performance is inseparable from implementation structure: lookup-table size, delimiter requirements, redundancy, auxiliary constraints, or the complexity of the search induced by the code. The resulting field is therefore best understood not as a single construction class, but as a family of observability designs in which code structure is introduced precisely to make position recoverable from limited or corrupted observations.