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Control-Field and Quasi-Cyclic Code Recovery in Starlink Downlink Signals

Haoyan Liu, Yichuan Ma, Min Yang, Chengguang Li

arXiv:2610.29387v1eess.SP

TL;DR

Prior Starlink descriptions establish synchronization, pilots, and predictable templates, but leave explicit control-field and payload-code representations unresolved. This paper recovers and tests control and coding relations in one continuous capture. Its conclusions are receiver-equivalent structural findings, with unresolved formats and limited evidence for generalization.

  • Problem

    Known synchronization and predictable waveform elements do not establish codeword boundaries, control-carrier locations, or whether remaining symbols follow an identified channel code.

    Source: §I · p. 1

  • Method

    The paper combines reference removal, control-field and allocation analysis, and quasi-cyclic algebraic recovery with sparse checks and reserved-word parity tests.

    Source: §I · p. 2, A. Coding Relations, QC Recovery, and… · p. 14

  • Results

    Recovered control and coding relations explain observed symbols, including full-rank sparse QC representations and shortened cores for codes 28 and 37, while code 4 remains unresolved.

    Source: §I · p. 2, §VII · p. 20

  • Takeaways & Limitations

    The recovered representations provide testable structural descriptions for this capture, rather than authenticated transmitter-standard identities or plaintext semantics.

    Source: §I · p. 2, §VII · p. 20

    Evidence is limited to one approximately 24 ms capture; two high-order formats have one target field each, and extra bits in codes 28 and 37 remain unexplained.

    Source: §VII · p. 20

Abstract

from arXiv · show

Public descriptions of the Starlink downlink specify synchronization sequences, edge pilots, and predictable symbol templates, but leave the coded control fields and explicit payload-code representations largely unresolved. We examine one continuous complex-baseband capture spanning approximately 24 ms and recover control and coding structure beyond those predictable elements. Reference-phase removal exposes terminated rate-1/3 convolutional control blocks with constraint length seven and a group-level cyclic redundancy check. A conditional descriptor layout predicts sequential allocations through per-codeword symbol rounding, agreeing exactly with independently estimated endpoints in 16 of 18 frame records and within two symbols in the remaining records. Quasi-cyclic closure, binary elimination, and sparse dual checks yield explicit systematic code representations at rates 1/2 and 5/8. Frozen encoders predict 78,848, 9,984, and 811,008 reserved parity bits for three formats without disagreement. A rank-deficient long-code fit is completed within a declared finite structural family, and shared encoders are tested across modulation formats. The recovered representations explain observed coded symbols; they do not authenticate transmitter-standard identities or plaintext semantics. Two high-order formats have only one target field each, while codes 28 and 37 admit verified shortened cores with unexplained extra bits and code 4 remains unresolved.

I. INTRODUCTION

The paper moves from known Starlink waveform features to explicit, testable control and coding relations, while treating the result as structural evidence for one capture rather than an authenticated standard identity. Its receiver analysis uses the known frame and pilot structure as a scaffold for that recovery.

I. INTRODUCTION

  • The open problem is to identify codeword boundaries, control carriers, and channel codes, which OFDM timing and predictable waveform elements alone do not specify.Prior work established frame and synchronization parameters, while later work recovered edge pilots and predictable templates.Source: §I · p. 1
  • The analysis separates deterministic reference-phase removal from data labeling before attempting algebraic code recovery, avoiding an unsupported inference about information-bit scrambling.It uses physically supported boundaries and tests ordered allocation and systematic/QC hypotheses rather than attempting exhaustive unconstrained protocol recovery.Source: §I · p. 2
  • Sparse QC checks, full-dimensional generator/check consistency, claimed ranks, and reserved-word parity prediction are required to support an LDPC-structure claim beyond a successful fit.Cyclic shifts supply equations only under a declared closure assumption and are not new independent transmissions.Source: §I · p. 2
  • Rank-deficient code-10 inputs leave encoder directions unobserved, so the fitting and verification scope is stated explicitly and pre-fixed observations are not treated as unseen tests.All available code-10 words lie in the same missing-direction hyperplane; agreement on another such word does not eliminate the ambiguity.Source: §I · p. 2
  • The conditional allocation layout and recovered control and coding relations provide an explicit, testable description beyond predictable waveform elements, not a general decoding algorithm or authenticated transmitter identity.The paper also tests frozen encoders through parity prediction and cross-modulation checks.Source: §I · p. 2

A. Signal-Format Review

  • The receiver analyzes 18 complete frames from one 24 ms baseband capture using a 302-slot frame grid, ordinary OFDM symbols, and 16 edge pilots.Nine narrowband tones outside the loaded OFDM band are tracked as a frequency-rate check, but unknown nominal tone frequencies prevent an absolute Doppler estimate.Source: A. Signal-Format Review · pp. 2–4

B. Received-Signal Equalization

The receiver corrects frequency and timing variation, then uses known synchronization and pilot structure for equalization. Tone-derived frequency-rate estimates complement pilot-anchored frequency estimates, while the residual checks apply to this capture rather than independent Doppler calibration.

B. Received-Signal Equalization

  • Eight of nine narrow tones pass the phase-fit test, yielding a rate estimate of −1107.157 Hz/s, while unknown tone baselines prevent an absolute frequency-offset estimate.The weak sixth tone is excluded because of phase-cycle errors.Source: B. Received-Signal Equalization · p. 4
  • The pilots estimate the remaining common frequency offset at the capture midpoint as 808.639 Hz and independently provide a −1165.205 Hz/s rate estimate for comparison with the tone-derived rate.Both estimates use the same capture.Source: B. Received-Signal Equalization · p. 4
  • 2.742306 Hz RMS residual remains after compensation, a check of the applied correction on the estimation capture rather than an independent Doppler calibration.The correction combines CP and pilot frequency estimates with the tone-derived rate.Source: B. Received-Signal Equalization · p. 4
  • Resampling on the estimated fractional timing grid reduces the illustrated frame’s relative timing RMS from 12.178511 to 0.052661 ns, and the equalized pilots check the receiver output.A phase trend fitted over slots 50–289 predicts early pilot labels in slots 2–16 with 0 errors among 3840 observations.Source: B. Received-Signal Equalization · p. 5

C. PN Phase Rotation

A compatible PN reference is aligned to the capture using known pilots and applied to non-pilot resources. After reference removal, the leading region supports a binary-label model that agrees with all tested parity observations.

C. PN Phase Rotation

  • The reference rotation is prior information from a compatible template, not a newly discovered transmitter operation.Qin et al. report predictable pilot coefficients and a reference template; the paper tests a compatible reference against this capture.Source: C. PN Phase Rotation · p. 5
  • The reference uses a PN15 sequence with a 32,767-bit nonzero-seed period and indices advanced by carrier and slot position.The saved alignment starts at slot 17; its two sequence indices at ordinal zero are 16,327 and 16,320.Source: C. PN Phase Rotation · p. 5
  • The saved pilot model predicts all 4,304 pilot labels over slots 33–301 without disagreement.Its alignment uses pilot components and training-symbol angular parity, with the initial values fixed from slots 17–31 and slot 32.Source: C. PN Phase Rotation · p. 5
  • 287,534 leading-region symbols across 18 frames satisfy the predicted parity, with reference-removed BPSK EVM of 0.1329–0.1536.In R02, the region contains 10,924 symbols; across the capture, its length ranges from 6,600 to 41,416.Source: C. PN Phase Rotation · p. 6
  • The reference-corrected real-part signs supply candidate coded bits, while the tested 114-symbol and 2,048-symbol lengths remain coding-block hypotheses until coding relations are checked.The leading-region lengths leave remainders of 456, 570, and 684 symbols when divided by 2,048, motivating a 114-symbol candidate interval.Source: C. PN Phase Rotation · p. 6

III. TERMINATED CONTROL GROUPS AND CODING SCHEMES

The paper tests a patent-motivated rate-1/3 convolutional-code family against reference-corrected candidate bits. Algebra identifies a three-output relation, and the recovered taps define a constraint-length-seven encoder.

A. Convolutional Generator Polynomials

  • The search tests a seven-tap, three-output convolutional model across offsets, prefix lengths, bit labels, and output orderings.The patent motivates the coding family, but the paper recovers rather than assumes the taps, block lengths, and termination in this capture.Source: A. Convolutional Generator Polynomials · pp. 6–7
  • The rate-1/3 family is tested because the cited patent describes a non-recursive, non-systematic header code with six one-bit memory registers and 64 states.The binary constellation alone would not determine the code rate.Source: A. Convolutional Generator Polynomials · p. 6
  • Pairwise output-stream equations eliminate the unknown common input, and solving them over GF(2) identifies candidate tap relations rather than decoded blocks.The test permits a fixed binary bias in each output-pair relation for the candidate polarity convention.Source: A. Convolutional Generator Polynomials · p. 7
  • Reference-corrected bits in triple-interleaved order retain a one-dimensional nullspace at prefix lengths 81–192, while other tested labels and orderings do not.At length 255 the relation no longer holds over the whole prefix, and several offsets separated by three bits retain the same relation.Source: A. Convolutional Generator Polynomials · p. 7
  • The recovered ordered octal generators are (155, 123, 137), defining a rate-1/3 encoder with constraint length seven.For a six-bit state v and input b, the stated register convention fixes tap direction, output order, and the next state.Source: A. Convolutional Generator Polynomials · p. 7

B. Convolutional Coding Blocks and Group Boundaries

Independent termination, tail, and re-encoding checks establish coding blocks and candidate block groups. A group-level CRC then predicts the terminal byte across all 36 groups in the 18 frames.

B. Convolutional Coding Blocks and Group Boundaries

  • Two adjacent 114-bit windows pass zero-state, six-zero-tail, and exact re-encoding checks, leaving 32 retained bits per block after tail removal.Each window corresponds to 38 input bits; the six zero inputs clear the encoder’s six-bit memory.Source: B. Convolutional Coding Blocks and… · p. 7
  • Frozen encoder settings reproduce all 3,420 coded symbols in first groups from the other 15 frames, with zero-state and six-zero-tail checks satisfied.The generator, polarity, output order, and state convention were frozen from R00, R02, and R03.Source: B. Convolutional Coding Blocks and… · p. 7

C. CRC Input and Polynomial

  • The CRC input concatenates 32 retained bits per block except the final eight-bit check value, yielding 56, 88, or 120 input bits for groups of two, three, or four blocks.Zero tails are excluded; any information-layer padding remains part of the CRC input.Source: C. CRC Input and Polynomial · p. 8
  • The tested degree-eight CRC polynomial is x^8 + x^6 + x^3 + x^2 + 1, with a full-group input rule and most-significant-bit-first convention.The final block’s check byte is itself convolutionally encoded.Source: C. CRC Input and Polynomial · p. 8
  • The CRC polynomial produces zero differences across all 36 groups, including changing contents and three group lengths beyond the fitting examples.The CRC parameters selected from six fitting groups are frozen for the remaining 30 groups in the other 15 frames.Source: C. CRC Input and Polynomial · pp. 8–9
  • Each frame has two CRC-checked control groups: a fixed 228-resource first group and a second group of two, three, or four coding blocks.The CRC supports grouping the retained bits; the later correspondence between group fields and following allocations supports their control interpretation.Source: C. CRC Input and Polynomial · p. 9

IV. CONTROL-FIELD INTERPRETATION AND RESOURCE ALLOCATION

Control-group prefixes support candidate role and count fields, while a conditional entry layout is tested against modulation and resource lengths. Per-codeword symbol rounding predicts a repeatable symbol restriction, but the layout and code parameters remain candidates.

A. Prefix Observations and a Candidate Entry Layout

  • All 18 first groups share the same first 16 bits and contain two coding blocks, while second groups show four prefix patterns and contain two, three, or four blocks.Bit 5 tracks group role, and bits 8–10 under the declared weighting convention match the independently recovered block count.Source: A. Prefix Observations and a Candidate… · p. 9
  • The proposed layout treats bits 12–15 as a candidate entry count D, distinct from block count N, followed by D equal-width entries, padding, and a known CRC.The layout is a hypothesis: constant prefixes and group-length associations alone do not establish the bit-16 boundary or prove that D counts allocations.Source: A. Prefix Observations and a Candidate… · p. 9

B. Entry-Field Interpretation from Modulation and Resource Lengths

  • Both 19-bit and 20-bit entry widths pass the zero-padding and minimum-32-bit-word tests across all 36 groups.The residual zeros could be padding or reserved fields, and CRC checks cannot distinguish the entry boundaries.Source: B. Entry-Field Interpretation from… · p. 9
  • Candidate six-bit format labels associate with independently compared modulation observations, including BPSK, QPSK, and sixteen-point constellations.The tested first entry begins at bit 16; its six bits are 101000 for R02’s first group, 010100 for its second, and 111001 for R17’s second.Source: B. Entry-Field Interpretation from… · p. 9
  • Measured BPSK intervals of 12,288, 10,240, and 6,144 symbols correspond to candidate first-group entry values L = 24, 20, and 12.The additional high bit is zero in these examples, so the 19-bit candidate gives the same numerical values as the 20-bit working layout.Source: B. Entry-Field Interpretation from… · p. 9
  • For R02’s code-28 candidate, per-codeword rounding predicts 273,100 symbols versus 273,067 under joint rounding, a 33-symbol difference.The candidate uses 50 codewords of 16,384 bits mapped at three bits per symbol; these are capacity parameters, not a recovered code-28 encoder.Source: B. Entry-Field Interpretation from… · pp. 10–11
  • The repeated last-symbol restriction supports separate final modulation symbols for codewords, but does not uniquely establish code length, bit labels, polarity, or spare-bit values.Under the candidate 16,384-bit model, a final symbol has capacity for two additional bits; analogous 8,192-bit candidates leave one bit of capacity.Source: B. Entry-Field Interpretation from… · p. 12

C. Payload-Endpoint Validation and Selection of the Entry Width

The conditional 20-bit descriptor layout is retained because its predicted allocation endpoints agree closely with signal-based transitions. The search distinguishes changing payload symbols from reference-symbol padding without using predicted endpoints as search constraints.

C. Payload-Endpoint Validation and Selection of the Entry Width

  • Reference-symbol padding is a received-power suffix that becomes concentrated near a constant after PN removal, unlike a silent guard.In R02, the changing payload ends at the start of region 8.Source: C. Payload-Endpoint Validation and… · p. 12
  • The signal-based endpoint search compares changing-symbol scores before a candidate transition with reference-symbol scores afterward, without constraining the transition to the descriptor prediction.It searches the whole interval for a single transition rather than selecting the first reference match.Source: C. Payload-Endpoint Validation and… · p. 12
  • 16 of 18 frame records match predicted and signal-based endpoints exactly; the other two differ by one and two symbols.For R02, 286071 predicted modulation symbols after the last control group reach the independently estimated transition.Source: C. Payload-Endpoint Validation and… · pp. 12–13
  • Both R04 and R11 show zero-symbol error at the measured internal boundary, while the comparison rejects the tested QPSK explanation without establishing a complete 16QAM mapping.The model scores and reference-removed constellation plots describe different processing stages.Source: C. Payload-Endpoint Validation and… · p. 13
  • Retaining the conditional 20-bit layout depends on ordered contiguous allocations, a common six-bit format convention, and consistent modulation meaning across positions.The endpoint search also depends on recovered reference and control-group position and tests a finite constellation bank with one terminal transition.Source: C. Payload-Endpoint Validation and… · pp. 12–13

V. RECOVERY OF EQUIVALENT QC CODE REPRESENTATIONS

The section recovers equivalent systematic generator and parity-check representations by combining binary elimination with declared quasi-cyclic structure. It tests these representations against reserved parity observations and signal-domain reconstructions, while limiting claims to the observed bit convention and structural assumptions.

V. RECOVERY OF EQUIVALENT QC CODE REPRESENTATIONS

  • When the systematic input matrix has full rank, consistency uniquely determines the parity-generation matrix in the chosen coordinate system; deficient rank leaves unobserved directions requiring data or structural assumptions.The fitted relation is UP = V, with P mapping systematic bits to parity bits over GF(2).Source: A. Coding Relations, QC Recovery, and… · p. 13
  • Agreement identifies a code relation under the observed bit convention, not the meaning of its systematic input.The tested allocations are independently identified before their algebraic coding relations are evaluated.Source: A. Coding Relations, QC Recovery, and… · pp. 13–14
  • Under a QC hypothesis, rotating each contiguous Z-bit block augments the equations, but the resulting identifiability holds only within the accepted closure model.These rotations are algebraic augmentation, not additional independently captured codewords.Source: A. Coding Relations, QC Recovery, and… · pp. 13–14
  • A complete sparse dual requires full-rank checks and consistency with the generator, rather than isolated low-weight equations or a dense fitted generator alone.When the parity part of H is nonsingular, H also provides a direct parity reconstruction independently of how P was stored.Source: A. Coding Relations, QC Recovery, and… · p. 14
  • Frozen parity-check matrices are tested through candidate bit labels, decoder diagnostics, re-encoded parity, and remodulation; zero decoder syndrome alone is insufficient identification evidence.The likelihood scale is a working assumption rather than a calibrated physical noise variance.Source: A. Coding Relations, QC Recovery, and… · p. 14
  • For four code-10 fields, frozen re-encoding and remodulation yields 0.01822–0.01922 NMSE, versus 1.97388–1.99058 for the parity-flip control.The comparison tests predicted redundancy while retaining the modulation alphabet; core-only and full-field denominators are not interchangeable.Source: A. Coding Relations, QC Recovery, and… · pp. 14–15

B. Code 5

Code 5 is recovered as a systematic relation for the first BPSK body, with sparse QC dual checks and successful prediction of reserved words. The evidence depends on QC structure and does not establish transmitter-standard identity or source semantics.

B. Code 5

  • Fourteen words augmented by QC128 rotations provide 1792 training equations and input rank 1024, yielding a 1024×1024 parity generator with zero fitted affine offset.Each 2048-bit unit predicts its following 1024 bits from the leading 1024 under the accepted receiver convention.Source: B. Code 5 · p. 15
  • Eight seed checks and their QC128 orbits form a rank-1024 sparse matrix with row weight six and column weights two, three, or four.The seeds are recovered by proposing small parity supports and verifying the associated systematic supports in the full dimension.Source: B. Code 5 · p. 15
  • 78,848 reserved parity bits across 77 first-body words from 15 other frame records disagree at zero positions under the frozen predictor.All 91 first-body words satisfy the recovered generator and checks; two later allocations with the same format code also pass.Source: B. Code 5 · p. 15
  • The 91 raw first-body words have rank at most 91, so identifying the 1024-dimensional representation relies on QC structure rather than independent acquisitions.Zero disagreement on reserved bits is not transmitter-ground-truth BER or 78,848 independent trials.Source: B. Code 5 · p. 15

C. Codes 10 and 39

The section tests frozen code relations across rank-deficient inputs and modulation formats, finding exact reserved-parity agreement within declared structural families while retaining limits on what those tests establish.

C. Codes 10 and 39

  • The code-10 fit has input rank 8,191 for a candidate information dimension of 8,192, leaving one input direction unobserved.All ten observed inputs lie in the same hyperplane; QC invariance leaves 256 completions, and the sparse prior is essential to selecting one within the declared family.Source: C. Codes 10 and 39 · p. 15
  • 8192 parity bits agree with the frozen code-10 encoder on F13, but that word is consistency evidence rather than an unseen-data test.The sparse search selects completion mask 31 only within its declared finite completion family and sparse-search procedure.Source: C. Codes 10 and 39 · p. 15
  • Code-10 completions remain ambiguous for inputs in the observed hyperplane; a word outside it would directly test the missing direction if the source permits one.If source inputs always obey the restriction, observations may identify only the transmitted subspace.Source: C. Codes 10 and 39 · p. 15
  • A 13,824-candidate mapping search for code 39 selects separable Gray labeling, but its best hard-decision candidate still has 59 failed checks.Only F17 contains the target code-39 field, and the next mapping candidate has more than 2,600 failed checks.Source: C. Codes 10 and 39 · p. 15
  • A frozen (8192, 4096) encoder predicted two code-3 words in F13 and one QPSK code-12 word in F16, with zero disagreements across 8,192 and 4,096 reserved parity bits, respectively.A separate code-12 refit reproduced the shifts but was not counted as a second independent validation.Source: D. Codes 3, 12, and 27 · p. 16
  • Code 27 passes all 4,096 checks and predicted parity bits after mapping selection on its sole target field.The complete 2,731-symbol remodulation assigns the final symbol's unassigned label bit to one; this fits the endpoint model but does not authenticate the transmitter's spare-bit value or confirm standard 8QAM.Source: D. Codes 3, 12, and 27 · p. 16

E. Code 9

The recovered codes are tested with frozen parity predictions across reserved words and, for code 37, against a shortened core of the code-21 encoder. These tests support specific code relations while leaving the extra code-37 bits unexplained.

E. Code 9

  • 9,984 reserved parity bits across 13 code-9 words are predicted without disagreement, and all 31 words pass.The recovered check matrix has rank 768 and row weights 7–16.Source: E. Code 9 · p. 16

F. Codes 21 and 37

  • 811,008 reserved parity bits across 264 code-21 words agree with the frozen encoder without disagreement.The recovered check matrix has rank 3,072 and row weights 7–18; the input-basis rank of 5,120 is not a claim about the rank of matrix P.Source: F. Codes 21 and 37 · p. 16
  • Code 21 is recovered independently because direct polynomial lifting from code 9 did not reproduce its observations.The failed transfer excludes that tested transformation, not every possible sharing transformation.Source: F. Codes 21 and 37 · p. 16
  • Code 37's final 3,072 observed bits agree with a shortened (7,808, 4,736) code-21 core, while 384 intervening bits remain unexplained.The first 384 mother-code input bits are fixed to zero, and 4,736 observed systematic bits occupy mother coordinates 384–5,119.Source: F. Codes 21 and 37 · p. 16
  • All seven code-37 words pass the shortened checks and re-encode at all 21,504 core parity coordinates.Core-only remodulation uses 2,601 predicted-symbol positions per word and gives NMSE 0.016956–0.026212; the remaining 130 symbols per word are omitted.Source: F. Codes 21 and 37 · p. 16
  • Fig. 12 shows equivalent rate-half QC-check incidence across code 5, codes 3/12, and code 10, with different lift sizes.The incidence pattern is shared, but shift magnitudes do not scale linearly with Z.Source: Figure 12 · p. 17

G. Verified Systematic and Parity Resources in R02

The R02 systematic/parity partitions permit frozen parity prediction, while code 28 is supported as a shortened core rather than a complete encoder. Extra observed bits remain unexplained.

G. Verified Systematic and Parity Resources in R02

  • Code 5's 2,048-symbol words split into 1,024 systematic and 1,024 parity symbols, while code 10's 16,384 bits split into two 4,096-symbol QPSK parts.The system/parity transition can lie inside an OFDM slot; Fig. 14 overlays the corrected unknown-field boundaries and coding partitions.Source: G. Verified Systematic and Parity… · p. 17
  • The code-5/code-10 overlay identifies recovered systematic and parity coordinates, but code-10 identification remains conditional on its finite completion family.The panels show these two partitions only; code 4 remains unresolved and codes 28 and 37 are evaluated separately.Source: G. Verified Systematic and Parity… · p. 17

H. Code 28

  • Code 28's QC1024 equations recover a (16,384, 9,216) mother encoder with nine systematic and seven parity cyclic blocks.Its sparse dual supplies 7,168 independent checks; the core fixes the first 256 mother-code input bits to zero and excludes the intervening observed bits.Source: H. Code 28 · p. 18
  • 637,952 reserved code-28 parity bits in R09/R13 are predicted with zero disagreement by the frozen encoder.The shortened (16,128, 8,960) core leaves 256 intervening observed bits unexplained; it does not recover a complete 16,384-bit encoder.Source: H. Code 28 · p. 18
  • All 1,354,752 core parity positions across 189 code-28 words agree with re-encoding, and all 189 words pass soft decoding.Core-only remodulation evaluates 5,375 of 5,462 symbols per word; its NMSE is 0.016995–0.022086, versus 1.5264–1.6310 for the parity-flip control.Source: H. Code 28 · p. 18
  • The extra 256 code-28 bits could be pseudorandom filling, but their generation rule and the serialization needed to reconcile their positions are unestablished.The patent describes appending bits to LDPC output, unlike the observed placement between systematic and parity segments.Source: H. Code 28 · p. 18

I. Code 4

The tested code-4 hypotheses fail at supported allocation boundaries, so its coding remains unidentified. The broader validation also limits what the single-capture waveform and coding results establish.

I. Code 4

  • The tested code-3 and code-21 checks and an arbitrary sixteen-point mapping search do not validate code 4 at the supported boundaries.The code-3 checks leave 1,989–2,068 of 4,096 equations nonzero, while code-21 checks leave 1,517–1,564 of 3,072 nonzero; these results do not reject all code-4 encoders.Source: §I · pp. 18–19
  • No code-4 generator, parity-check matrix, information dimension, or rate is identified.Its field position and sixteen-point observations remain supported separately from its unresolved coding.Source: §I · p. 19

J. Validation Scope and Waveform Reconstruction

  • Across 16 reconstructed frames, full-period NMSE is 0.01860–0.02499 and data-region NMSE is 0.01842–0.02482.The training residual is a same-symbol fit, not independent channel validation; the low residuals support waveform consistency without requiring recovery of all FEC.Source: J. Validation Scope and Waveform… · p. 19
  • The separate frozen parity predictions provide stronger coding evidence than waveform remodulation, which does not prove unresolved encoders.The manuscript also states that code 4 remains unresolved and codes 28/37 have shortened-core rather than complete-field recovery.Source: J. Validation Scope and Waveform… · p. 20
  • All 18 records come from one continuous capture, and no post-fit new capture or independent bit trials are included.Codes 39 and 27 each have only one target-format field, and zero parity disagreement is measured against recovered receiver bits rather than transmitted truth.Source: J. Validation Scope and Waveform… · pp. 19–20

VI. COMPARISON WITH PUBLISHED MODULATION AND CODING CONFIGURATIONS

The comparison finds numerical compatibility with some published example configurations, but not authenticated encoder or constellation identity. The conclusion also records unresolved code boundaries and limits the findings to the observed capture.

VI. COMPARISON WITH PUBLISHED MODULATION AND CODING CONFIGURATIONS

  • Patent rows are examples from a stated subset, so a missing row provides no direct comparison and does not show that a format is absent.The comparison is made after receiver labels and coding relations are established, not to redefine the control-field bit split.Source: §VI · p. 20
  • The control row encodes 38 input clocks, including six termination clocks, into 114 transmitted bits; its 32 retained bits include a group CRC contribution.The patent’s K = 32 does not imply an effective retained-bit ratio of 1/3 for those 114 bits.Source: §VI · p. 20
  • Codes 27 and 28 occupy eight-point subsets of a sixteen-point grid, but their unequal magnitudes do not establish the example rows’ 8PSK constellation identity.Code 28’s 8960 information coordinates over 16384 nominal positions give 35/64, while its verified core rate is 8960/16128 = 5/9; code 37’s verified core rate is 4736/7808 = 37/61, and extra bits remain unexplained.Source: §VI · p. 20
  • Patent comparisons establish compatible parameters or candidate mechanisms, not parity-check-matrix equality or authenticated transmitter-standard identities.Equivalent basis changes, receiver labeling, and phase conventions can change the displayed representation without changing the observable code space.Source: §VI · p. 20

VII. CONCLUSION

  • The capture supports receiver-equivalent code relations, but codes 28 and 37 retain unexplained extra bits, code 4 remains unresolved, and further acquisitions are needed to test generalization.Patent comparisons agree with several observed lengths and rates, but do not establish complete encoder identity; the observed eight-point alphabets differ from the example 8PSK labels.Source: §VII · p. 20
  • Appendix panels map complete frame regions and reference-removed constellations, with conditional status retained for code-4/28/37 boundaries.R07 and R16 retain descriptor-minus-signal endpoint differences of one and two symbols; displayed constellations use up to 4000 actual observations each.Source: Appendix A · pp. 20–21, Figure 15 · p. 22, Figure 16 · p. 23

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