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arxiv: 2604.18335 · v1 · submitted 2026-04-20 · 💻 cs.IT · math.IT

Polar Coded Quantization for Distributed Source Coding

Muhammed Yusuf Sener , Gerhard Kramer , Shlomo Shamai (Shitz) , Ronald B\"ohnke , Wen Xu This is my paper

Pith reviewed 2026-05-10 03:37 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords polar coded quantizationdistributed source codingBerger-Tung regionWyner-Ziv codingGaussian sourcesmean-square errorditheringtruncated Gaussian shaping
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The pith

Polar coded quantization with dithering and truncated Gaussian shaping achieves the corner points of the Berger-Tung region for Gaussian sources.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a quantization scheme for two correlated Gaussian sources encoded separately but reconstructed jointly. It places scalar quantization inside a modulo interval, adds dithering, and applies truncated Gaussian shaping to reach the optimal performance limits at the corner points of the Berger-Tung region. The construction is realized with short-block-length multilevel 5G polar codes in the Wyner-Ziv setting. The joint scheme produces lower total distortion than separate quantization of each source block. This matters for sensor networks and other systems that must compress correlated observations without exchanging full side information.

Core claim

The paper shows that a coding scheme combining scalar quantization, a modulo interval, dithering, and truncated Gaussian shaping, when implemented with polar codes, attains the corner points of the Berger-Tung region for jointly Gaussian sources under mean-square error distortion. Design of short-block-length 5G polar codes for Wyner-Ziv polar coded quantization illustrates that the approach yields substantially lower total distortion than independent quantization of the separate source blocks.

What carries the argument

The polar coded quantization scheme that places quantization inside a modulo interval together with dithering and truncated Gaussian shaping.

If this is right

  • Short polar codes become usable for Wyner-Ziv coding while meeting known theoretical limits at the corner points.
  • Total distortion falls when the decoder exploits correlation between the sources rather than quantizing them independently.
  • Polar codes gain an additional role in source coding beyond their established use in channel coding.
  • Corner-point achievability supplies a concrete step toward constructing codes for the entire Berger-Tung region.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same dithering and shaping construction could be tested on sources that are not jointly Gaussian by substituting appropriate shaping distributions.
  • Explicit bounds on the gap caused by finite-length effects would quantify how close practical codes come to the ideal corner points.
  • The modular structure may extend to coding schemes for more than two terminals that share correlated observations.

Load-bearing premise

The sources are jointly Gaussian, distortion is measured by mean-square error, and ideal dither plus shaping exist for the finite-length polar codes used.

What would settle it

A rate-distortion measurement on finite-length polar codes that yields a total distortion strictly larger than the value predicted at the Berger-Tung corner points.

Figures

Figures reproduced from arXiv: 2604.18335 by Gerhard Kramer, Muhammed Yusuf Sener, Ronald B\"ohnke, Shlomo Shamai (Shitz), Wen Xu.

Figure 1
Figure 1. Figure 1: WZ coding with a modulo operator and dithering. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Distortion region for a Gaussian source with covariance matrix (27) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pr [∆ > D] for the source σ 2 x1 = σ 2 x2 = 2.5 and ρ = 0.8, the rates R1 = 1 and R2 = 2, and with 8-ASK and 16-ASK for the first and second source, respectively. We optimize these parameters by targeting the ideal rates of Sec. III-B but back off from the ideal distortions [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

Scalar quantization and probabilistic shaping are applied to the distributed source coding of Gaussian sources, with mean-square error distortion. A coding scheme with a modulo interval, dithering, and truncated Gaussian shaping is shown to achieve the corner points of the Berger-Tung region. The theory is illustrated by designing short-block-length multilevel 5G polar codes for Wyner-Ziv (WZ) polar coded quantization (PCQ). WZ-PCQ substantially reduces the total distortion compared to separate PCQ of the source blocks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims to develop a polar coded quantization (PCQ) approach for distributed source coding of jointly Gaussian sources with mean-square error (MSE) distortion. By incorporating a modulo interval, dithering, and truncated Gaussian shaping into the quantization process, the scheme is asserted to achieve the corner points of the Berger-Tung achievable region. The authors further illustrate this with the design of short-block-length multilevel 5G polar codes applied to the Wyner-Ziv (WZ) setting, demonstrating that WZ-PCQ substantially lowers the total distortion relative to performing PCQ separately on each source block.

Significance. If the finite-length construction with truncation and polar codes indeed attains the corner points without rate-distortion loss, the work would provide a practical, implementable method to realize information-theoretic limits for distributed Gaussian source coding. This could impact applications in sensor networks and correlated data compression by offering low-complexity polar-code-based solutions.

major comments (2)
  1. [Abstract] Abstract: The assertion that the scheme 'is shown to achieve' the corner points of the Berger-Tung region relies on idealized dither and shaping. However, truncation of the Gaussian shaping density creates a nonzero total-variation distance to the ideal auxiliary, and finite-length multilevel polar codes under successive cancellation have positive block-error probability. These must be shown not to induce a strict gap; a quantitative error analysis or convergence theorem is required to support exact achievability.
  2. [WZ-PCQ construction] WZ-PCQ construction and numerical illustration: The short-block-length 5G polar code examples should explicitly compare the achieved distortion to the theoretical Berger-Tung corner-point value for the same rate, to demonstrate whether the finite-length effects close the gap or leave a measurable loss relative to the claimed achievement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments and the recommendation for major revision. We have carefully considered the points raised and provide point-by-point responses below, along with our plans for revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The assertion that the scheme 'is shown to achieve' the corner points of the Berger-Tung region relies on idealized dither and shaping. However, truncation of the Gaussian shaping density creates a nonzero total-variation distance to the ideal auxiliary, and finite-length multilevel polar codes under successive cancellation have positive block-error probability. These must be shown not to induce a strict gap; a quantitative error analysis or convergence theorem is required to support exact achievability.

    Authors: We acknowledge that the use of truncated Gaussian shaping introduces a nonzero total-variation distance compared to the ideal Gaussian auxiliary distribution, and that finite-length polar codes exhibit a positive error probability under successive cancellation decoding. Our original analysis establishes achievability assuming idealized dithering and shaping, with the truncated version presented as a practical realization. To rigorously demonstrate that these practical aspects do not create a strict gap to the Berger-Tung corner points, we will incorporate a quantitative error analysis in the revised manuscript. This analysis will bound the excess distortion due to truncation (by controlling the truncation threshold) and due to decoding errors (by the vanishing error probability of polar codes at rates below capacity), showing convergence to the target distortion as the block length and truncation range increase. revision: yes

  2. Referee: [WZ-PCQ construction] WZ-PCQ construction and numerical illustration: The short-block-length 5G polar code examples should explicitly compare the achieved distortion to the theoretical Berger-Tung corner-point value for the same rate, to demonstrate whether the finite-length effects close the gap or leave a measurable loss relative to the claimed achievement.

    Authors: We agree that explicitly comparing the simulated distortions to the theoretical Berger-Tung corner-point distortions would better illustrate the performance gap due to finite block lengths. In the revised manuscript, we will add such comparisons for the presented 5G polar code examples, including the theoretical distortion values computed from the Berger-Tung region for the operating rates, to quantify the loss. revision: yes

Circularity Check

0 steps flagged

No significant circularity; achievability rests on standard Berger-Tung and polar-code results

full rationale

The paper claims that a modulo-interval + dither + truncated-Gaussian-shaping construction paired with multilevel polar codes achieves the corner points of the Berger-Tung region for jointly Gaussian sources under MSE. This is an existence-style achievability argument that invokes the known characterization of the Berger-Tung region and standard polar-code constructions for quantization. No equation or step in the abstract reduces a claimed prediction to a fitted parameter defined by the authors themselves, nor does any load-bearing premise collapse to a self-citation whose content is itself unverified within the paper. The derivation therefore remains self-contained against external benchmarks (the Berger-Tung theorem and polar-code literature) and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard assumption that the sources are jointly Gaussian and that mean-square error is the distortion measure; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Sources are jointly Gaussian with mean-square error distortion
    Invoked to define the Berger-Tung region and the target corner points.

pith-pipeline@v0.9.0 · 5383 in / 1180 out tokens · 33098 ms · 2026-05-10T03:37:50.397647+00:00 · methodology

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Channel polarization: A method for constructing capacity- achieving codes for symmetric binary-input memoryless channels,

    E. Arikan, “Channel polarization: A method for constructing capacity- achieving codes for symmetric binary-input memoryless channels,”IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, 2009

    work page 2009

  2. [2]

    Writing on dirty paper,

    M. Costa, “Writing on dirty paper,”IEEE Trans. Inf. Theory, vol. 29, no. 3, pp. 439–441, 1983

    work page 1983

  3. [3]

    The rate-distortion function for source coding with side information at the decoder,

    A. Wyner and J. Ziv, “The rate-distortion function for source coding with side information at the decoder,”IEEE Trans. Inf. Theory, vol. 22, no. 1, pp. 1–10, 1976

    work page 1976

  4. [4]

    Polar codes are optimal for lossy source coding,

    S. B. Korada and R. L. Urbanke, “Polar codes are optimal for lossy source coding,”IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1751–1768, 2010

    work page 2010

  5. [5]

    Achieving the capacity of any DMC using only polar codes,

    D. Sutter, J. M. Renes, F. Dupuis, and R. Renner, “Achieving the capacity of any DMC using only polar codes,” inIEEE Inf. Theory Workshop, Lausanne, Switzerland, 2012, pp. 114–118

    work page 2012

  6. [6]

    Polar coding without alphabet extension for asymmetric models,

    J. Honda and H. Yamamoto, “Polar coding without alphabet extension for asymmetric models,”IEEE Trans. Inf. Theory, vol. 59, no. 12, pp. 7829–7838, 2013

    work page 2013

  7. [7]

    Polar-coded modulation,

    M. Seidl, A. Schenk, C. Stierstorfer, and J. B. Huber, “Polar-coded modulation,”IEEE Trans. Commun., vol. 61, no. 10, pp. 4108–4119, 2013

    work page 2013

  8. [8]

    A new multilevel coding method using error- correcting codes,

    H. Imai and S. Hirakawa, “A new multilevel coding method using error- correcting codes,”IEEE Trans. Inf. Theory, vol. 23, no. 3, pp. 371–377, 1977

    work page 1977

  9. [9]

    Multilevel codes: theoretical concepts and practical design rules,

    U. Wachsmann, R. F. Fischer, and J. B. Huber, “Multilevel codes: theoretical concepts and practical design rules,”IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1361–1391, 1999

    work page 1999

  10. [10]

    Efficient polar code construction for higher-order modulation,

    G. B ¨ocherer, T. Prinz, P. Yuan, and F. Steiner, “Efficient polar code construction for higher-order modulation,” inIEEE Wireless Commun. Networking Conf. Workshops, San Francisco, CA, USA, 2017, pp. 1–6

    work page 2017

  11. [11]

    Polar coded probabilistic amplitude shaping for short packets,

    T. Prinz, P. Yuan, G. B ¨ocherer, F. Steiner, O. ˙Is ¸can, R. B¨ohnke, and W. Xu, “Polar coded probabilistic amplitude shaping for short packets,” inIEEE Int. Workshop Signal Proc. Advances in Wireless Commun., Sapporo, Japan, 2017, pp. 1–5

    work page 2017

  12. [12]

    Polar codes for a quadratic- Gaussian Wyner-Ziv problem,

    S. Eghbalian-Arani and H. Behroozi, “Polar codes for a quadratic- Gaussian Wyner-Ziv problem,” inInt. Symp. Wireless Commun. Sys., Ilmenau, Germany, 2013, pp. 1–5

    work page 2013

  13. [13]

    Polar codes and polar lattices for independent fading channels,

    L. Liu and C. Ling, “Polar codes and polar lattices for independent fading channels,”IEEE Trans. Commun., vol. 64, no. 12, pp. 4923–4935, 2016

    work page 2016

  14. [14]

    Construction of capacity-achieving lattice codes: Polar lattices,

    L. Liu, Y . Yan, C. Ling, and X. Wu, “Construction of capacity-achieving lattice codes: Polar lattices,”IEEE Trans. Commun., vol. 67, no. 2, pp. 915–928, 2019

    work page 2019

  15. [15]

    Polar lattices for lossy compression,

    L. Liu, J. Shi, and C. Ling, “Polar lattices for lossy compression,”IEEE Trans. Inf. Theory, vol. 67, no. 9, pp. 6140–6163, 2021

    work page 2021

  16. [16]

    On the equivalence between probabilistic shaping and geometric shaping: A polar lattice perspective,

    L. Liu, S. Lyu, C. Ling, and B. Bai, “On the equivalence between probabilistic shaping and geometric shaping: A polar lattice perspective,” inIEEE Int. Symp. Inf. Theory, Athens, Greece, 2024, pp. 2174–2179

    work page 2024

  17. [17]

    Universal Gaussian quantization with side-information using polar lattices,

    S. Jha, “Universal Gaussian quantization with side-information using polar lattices,”IEEE J. Sel. Areas Inf. Theory, vol. 3, no. 4, pp. 639–650, 2022

    work page 2022

  18. [18]

    How to achieve the capacity of asymmetric channels,

    M. Mondelli, S. H. Hassani, and R. L. Urbanke, “How to achieve the capacity of asymmetric channels,”IEEE Trans. Inf. Theory, vol. 64, no. 5, pp. 3371–3393, 2018

    work page 2018

  19. [19]

    Shaped polar codes for higher order modulation,

    O. ˙Is ¸can, R. B¨ohnke, and W. Xu, “Shaped polar codes for higher order modulation,”IEEE Commun. Lett., vol. 22, no. 2, pp. 252–255, 2018

    work page 2018

  20. [20]

    Probabilistic shaping using 5G New Radio polar codes,

    ——, “Probabilistic shaping using 5G New Radio polar codes,”IEEE Access, vol. 7, pp. 22 579–22 587, 2019

    work page 2019

  21. [21]

    Sign-bit shaping using polar codes,

    ——, “Sign-bit shaping using polar codes,”Trans. Emerging Telecom- mun. Technol., vol. 31, no. 10, p. e4058, 2020

    work page 2020

  22. [22]

    Shaped on-off keying using polar codes,

    T. Wiegart, F. Steiner, P. Schulte, and P. Yuan, “Shaped on-off keying using polar codes,”IEEE Commun. Lett., vol. 23, no. 11, pp. 1922–1926, 2019

    work page 1922

  23. [23]

    Multi-level distribution matching,

    R. B ¨ohnke, O. ˙Is ¸can, and W. Xu, “Multi-level distribution matching,” IEEE Commun. Lett., vol. 24, no. 9, pp. 2015–2019, 2020

    work page 2015

  24. [24]

    Multilevel binary polar- coded modulation achieving the capacity of asymmetric channels,

    C. Runge, T. Wiegart, D. Lentner, and T. Prinz, “Multilevel binary polar- coded modulation achieving the capacity of asymmetric channels,” in IEEE Int. Symp. Inf. Theory, Espoo,Finland, 2022, pp. 2595–2600

    work page 2022

  25. [25]

    Time-shifted alternating Gelfand-Pinsker coding for broadcast channels,

    C. Runge and G. Kramer, “Time-shifted alternating Gelfand-Pinsker coding for broadcast channels,” inIEEE Int. Symp. Inf. Theory, Athens, Greece, 2024, pp. 1700–1705

    work page 2024

  26. [26]

    Dirty paper coding based on polar codes and probabilistic shaping,

    M. Y . S ¸ener, R. B¨ohnke, W. Xu, and G. Kramer, “Dirty paper coding based on polar codes and probabilistic shaping,”IEEE Commun. Lett., vol. 25, no. 12, pp. 3810–3813, 2021

    work page 2021

  27. [27]

    Achieving the dirty paper channel capacity with scalar lattices and probabilistic shaping,

    ——, “Achieving the dirty paper channel capacity with scalar lattices and probabilistic shaping,”IEEE Commun. Lett., vol. 28, no. 1, pp. 29–33, 2024

    work page 2024

  28. [28]

    Achieving Gaussian vector broadcast channel capacity with scalar lattices,

    M. Y . S ¸ener, G. Kramer, S. Shamai Shitz, R. B ¨ohnke, and W. Xu, “Achieving Gaussian vector broadcast channel capacity with scalar lattices,” inIEE Int. Symp. Inf. Theory, Athens, Greece, 2024, pp. 1706– 1711

    work page 2024

  29. [29]

    Scalar lattices and probabilistic shaping for dithered wyner-ziv quantization,

    M. Y . S ¸ener, G. Kramer, S. S. Shitz, and W. Xu, “Scalar lattices and probabilistic shaping for dithered wyner-ziv quantization,” inIEE Int. Symp. Inf. Theory, Ann Arbor, MI, USA, 2025, pp. 1–6

    work page 2025

  30. [30]

    Nested linear/lattice codes for Wyner-Ziv encoding,

    R. Zamir and S. Shamai, “Nested linear/lattice codes for Wyner-Ziv encoding,” inInf. Theory Workshop, Killarney, Ireland, 1998, pp. 92– 93

    work page 1998

  31. [31]

    Nested linear/lattice codes for structured multiterminal binning,

    R. Zamir, S. Shamai, and U. Erez, “Nested linear/lattice codes for structured multiterminal binning,”IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1250–1276, 2002

    work page 2002

  32. [32]

    Zamir,Lattice Coding for Signals and Networks

    R. Zamir,Lattice Coding for Signals and Networks. Cambridge, U.K.: Cambridge Univ. Press, 2014

    work page 2014

  33. [33]

    AWGN-goodness is enough: Capacity-achieving lattice codes based on dithered probabilistic shaping,

    A. Campello, D. Dadush, and C. Ling, “AWGN-goodness is enough: Capacity-achieving lattice codes based on dithered probabilistic shaping,” IEEE Trans. Inf. Theory, vol. 65, no. 3, pp. 1961–1971, 2019

    work page 1961

  34. [34]

    Nested lattice coding with algebraic encoding and geometric decoding,

    H. Dongbo, H. Gang, X. Yonggang, and Y . Hongsheng, “Nested lattice coding with algebraic encoding and geometric decoding,”IEEE Access, vol. 9, pp. 11 598–11 609, 2021

    work page 2021

  35. [35]

    Multiterminal source coding,

    T. Berger, “Multiterminal source coding,” inThe Information Theory Ap- proach to Communications, ser . CISM Courses and Lectures, G. Longo, Ed. Vienna/New York: Springer Verlag, 1978, p. 171–231

    work page 1978

  36. [36]

    Multiterminal source coding,

    S. Y . Tung, “Multiterminal source coding,” Ph.D. dissertation, School of Elect. Eng., Cornell Univ., Ithaca, NY , USA, 1978

    work page 1978

  37. [37]

    An upper bound on the rate distortion function for source coding with partial side information at the decoder,

    T. Berger, K. Housewright, J. Omura, S. Yung, and J. Wolfowitz, “An upper bound on the rate distortion function for source coding with partial side information at the decoder,”IEEE Trans. Inf. Theory, vol. 25, no. 6, pp. 664–666, 1979

    work page 1979

  38. [38]

    Rate region of the quadratic Gaussian two-encoder source-coding problem,

    A. B. Wagner, S. Tavildar, and P. Viswanath, “Rate region of the quadratic Gaussian two-encoder source-coding problem,”IEEE Trans. Inf. Theory, vol. 54, no. 5, pp. 1938–1961, 2008

    work page 1938

  39. [39]

    The CEO problem [multi- terminal source coding],

    T. Berger, Z. Zhang, and H. Viswanathan, “The CEO problem [multi- terminal source coding],”IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 887–902, 1996

    work page 1996

  40. [40]

    Rate-distortion theory for Gaussian multiterminal source coding systems with several side informations at the decoder,

    Y . Oohama, “Rate-distortion theory for Gaussian multiterminal source coding systems with several side informations at the decoder,”IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2577–2593, 2005

    work page 2005

  41. [41]

    Rate region of the quadratic Gaussian CEO problem,

    V . Prabhakaran, D. Tse, and K. Ramachandran, “Rate region of the quadratic Gaussian CEO problem,” inIEEE Int. Symp. Inf. Theory. Chicago, IL: IEEE, 2004, p. 119

    work page 2004

  42. [42]

    Channel coding techniques for communication over net- works and over channels with memory,

    N. Ghaddar, “Channel coding techniques for communication over net- works and over channels with memory,” Ph.D. dissertation, University of California, San Diego, 2022

    work page 2022

  43. [43]

    A. E. Gamal and Y .-H. Kim,Network Information Theory. Cambridge Univ. Press, 2011

    work page 2011

  44. [44]

    Successive Wyner–Ziv coding scheme and its application to the quadratic Gaussian CEO problem,

    J. Chen and T. Berger, “Successive Wyner–Ziv coding scheme and its application to the quadratic Gaussian CEO problem,”IEEE Trans. Inf. Theory, vol. 54, no. 4, pp. 1586–1603, 2008

    work page 2008

  45. [45]

    Information theory for cellular wireless networks,

    Y .-H. Kim and G. Kramer, “Information theory for cellular wireless networks,” inInformation Theoretic Perspectives on 5G Systems and Beyond, I. Mari ´c, S. Shamai, and O. Simeone, Eds. Cambridge Univ. Press, 2022, p. 10–92

    work page 2022

  46. [46]

    Successive cancellation list decoding of BMERA codes with application to higher-order modulation,

    T. Prinz and P. Yuan, “Successive cancellation list decoding of BMERA codes with application to higher-order modulation,” inIEEE Int. Symp. Turbo Codes Iter . Inf. Proc., Hong Kong, China, 2018, pp. 1–5

    work page 2018

  47. [47]

    Joint list multistage decoding with sphere detection for polar coded SCMA systems,

    L. Karakchieva and P. Trifonov, “Joint list multistage decoding with sphere detection for polar coded SCMA systems,” inInt. ITG Conf. Sys., Commun. Coding, Rostock, Germany, 2019, pp. 1–6

    work page 2019