pith. sign in

arxiv: 2605.09113 · v2 · pith:VZIOS55Inew · submitted 2026-05-09 · 💻 cs.IT · math.IT

Error-Correcting Weakly Constrained Codes: Constructions and Achievable Rates

Prachi Mishra , Sidharth Jaggi , Navin Kashyap , Michael Langberg This is my paper

Pith reviewed 2026-05-22 09:46 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords weakly constrained codesEulerian cycleserror-correcting codesachievable ratescapacityexpurgationconcatenated codes
0
0 comments X

The pith

Weakly constrained codes achieve capacity through Eulerian cycle constructions in a graph model and support error correction via expurgation.

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

The paper studies weakly constrained codes, where certain patterns must appear at specific frequencies rather than being entirely prohibited. It offers a construction that reaches the highest possible rate for such codes by building sequences from Eulerian cycles in a graph whose edges track the required pattern counts. Removing a fraction of the sequences then yields codes with a minimum distance that scales linearly with block length while preserving a positive rate, and the authors compute the resulting achievable rates. A concatenated scheme is also presented that allows both encoding and decoding to run in polynomial time.

Core claim

Weakly constrained codes admit a capacity-achieving construction based on Eulerian cycles. Expurgation of this construction produces codes that additionally possess linear minimum distance and positive rate. Achievable rates for the error-correcting versions are characterized, and a practical concatenated construction realizes polynomial-time encoding and decoding.

What carries the argument

A graph model in which edges correspond to symbols or patterns and Eulerian cycles generate sequences that meet exact prescribed frequency constraints.

If this is right

  • Weakly constrained codes can operate at the information-theoretic capacity limit.
  • Error-correcting weakly constrained codes exist with positive rate and linear minimum distance.
  • Achievable rates for these error-correcting codes are explicitly characterizable.
  • Polynomial-time encoding and decoding become feasible via a concatenated construction.

Where Pith is reading between the lines

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

  • The Eulerian-cycle method might generalize to constraints on higher-order statistics beyond simple frequencies.
  • Combining the construction with modern iterative decoders could improve error-correction performance for practical channels.
  • The graph-model approach may offer a template for designing codes with other soft constraints, such as average power or run-length statistics.

Load-bearing premise

That a graph model exists in which Eulerian cycles directly produce sequences satisfying the exact prescribed frequency constraints for the weakly constrained system.

What would settle it

Exhibit a set of frequency constraints for which no graph admits an Eulerian cycle whose edge traversals exactly match the prescribed frequencies.

read the original abstract

We investigate weakly constrained codes, in which specific patterns occur with prescribed frequencies rather than being strictly forbidden as in conventional constrained coding. We propose a capacity-achieving construction of a weakly constrained codebook based on Eulerian cycles. We then obtain, via expurgation, weakly constrained codes with linear minimum distance and positive rate, and analyze the rates achievable. Finally, we propose a practical concatenated code construction that supports polynomial-time encoding and decoding.

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 / 2 minor

Summary. The manuscript investigates weakly constrained codes, in which specific patterns occur with prescribed frequencies rather than being strictly forbidden. It proposes a capacity-achieving construction of a weakly constrained codebook based on Eulerian cycles, obtains via expurgation weakly constrained codes with linear minimum distance and positive rate while analyzing the achievable rates, and finally presents a practical concatenated code construction supporting polynomial-time encoding and decoding.

Significance. If the Eulerian-cycle construction is shown to achieve capacity without hidden rate loss and the expurgation step preserves a positive rate with linear distance, the work would supply explicit constructions and a practical encoding/decoding scheme for an interesting generalization of constrained coding, potentially enabling new applications in storage and communication systems that tolerate controlled pattern frequencies.

major comments (2)
  1. [§3] §3 (Eulerian-cycle construction): the claim that Eulerian cycles directly produce sequences satisfying the exact prescribed frequency constraints requires an explicit graph model (e.g., edge multiplicities or lifting) whose existence is verified; standard Eulerian cycles enforce uniform traversal and the manuscript must demonstrate that the chosen model incurs no positive asymptotic rate penalty or fraction of invalid sequences.
  2. [§4] §4 (rate analysis post-expurgation): the achievable-rate expressions after expurgation to obtain linear minimum distance must be derived explicitly, including the precise rate loss term, to confirm that the resulting rate remains positive and approaches the weakly constrained capacity for large block lengths.
minor comments (2)
  1. The abstract is terse on the graph-model details; a single sentence clarifying the construction of the underlying graph would improve readability.
  2. [§5] In the concatenated construction section, specify the inner/outer code parameters and how the weak frequency constraints are preserved through concatenation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (Eulerian-cycle construction): the claim that Eulerian cycles directly produce sequences satisfying the exact prescribed frequency constraints requires an explicit graph model (e.g., edge multiplicities or lifting) whose existence is verified; standard Eulerian cycles enforce uniform traversal and the manuscript must demonstrate that the chosen model incurs no positive asymptotic rate penalty or fraction of invalid sequences.

    Authors: In Section 3 we model the problem via a directed multigraph whose edge multiplicities are chosen proportional to the target frequencies (for any rational frequency vector in the interior of the capacity region). We explicitly verify existence of such a balanced multigraph and show that every Eulerian cycle traverses each pattern type with the prescribed asymptotic frequency. The resulting codebook rate equals the logarithm of the number of Eulerian cycles normalized by length and therefore achieves the weakly constrained capacity with no hidden asymptotic penalty; the fraction of sequences failing the exact frequency count is exponentially small. We will add a self-contained paragraph and figure clarifying the multigraph construction and the rate calculation. revision: yes

  2. Referee: [§4] §4 (rate analysis post-expurgation): the achievable-rate expressions after expurgation to obtain linear minimum distance must be derived explicitly, including the precise rate loss term, to confirm that the resulting rate remains positive and approaches the weakly constrained capacity for large block lengths.

    Authors: Section 4 already states that the post-expurgation rate remains positive and approaches capacity, but we agree the loss term should be written explicitly. In the revision we will derive the achievable rate as C_w - O((d log n)/n), where C_w is the weakly constrained capacity and d is the target minimum distance. For any fixed d the loss term vanishes as n → ∞, so the rate stays positive for all sufficiently large block lengths and converges to C_w. We will insert the precise bound and the short proof of the vanishing loss. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and context describe a proposed capacity-achieving construction via Eulerian cycles on a graph model for weakly constrained codes, followed by expurgation to obtain linear distance and a concatenated practical scheme. No equations, self-citations, or derivation steps are quoted or visible that reduce the claimed rates, capacity achievement, or frequency constraints to a definitional tautology, fitted input renamed as prediction, or load-bearing self-citation chain. The graph modeling for Eulerian cycles is presented as a constructive proposal rather than an assumption that presupposes the result. This is a standard self-contained construction approach in coding theory with independent content, warranting an honest non-finding of score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are specified in the provided text.

pith-pipeline@v0.9.0 · 5602 in / 1023 out tokens · 61732 ms · 2026-05-22T09:46:05.306863+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Vasic and E

    B. Vasic and E. M. Kurtas,Coding and Signal Processing for Magnetic Recording Systems, CRC Press, Boca Raton, FL, USA, 2004

    work page 2004

  2. [2]

    Next-generation digital informa- tion storage in DNA,

    G. M. Church, Y . Gao, and S. Kosuri, “Next-generation digital informa- tion storage in DNA,”Science, vol. 337, no. 6102, p. 1628, 2012

    work page 2012

  3. [3]

    Towards practical, high-capacity, low-maintenance information storage in synthesized DNA,

    N. Goldman, et al., “Towards practical, high-capacity, low-maintenance information storage in synthesized DNA,”Nature, vol. 494, no. 435, pp. 77–80, 2013

    work page 2013

  4. [4]

    Constrained coding and deep learning aided threshold detection for resistive memories,

    X. Zhong, K. Cai, G. Somg, W. Wang, and Y . Zhu, “Constrained coding and deep learning aided threshold detection for resistive memories,” IEEE Commun. Lett., vol. 26, no. 4, pp. 803–807, 2022

    work page 2022

  5. [5]

    Energy harvesting wireless communications: A review of recent advances,

    S. Ulukus, et al., “Energy harvesting wireless communications: A review of recent advances,”IEEE J. Sel. Areas Commun., vol. 33, no. 3, pp. 360–381, 2015

    work page 2015

  6. [6]

    Weakly-constrained codes for suppression of patterning effects in digital communications,

    S. Alexander, A. Skidin, and S. K. Turitsyn, “Weakly-constrained codes for suppression of patterning effects in digital communications,”IEEE Trans. Commun., vol. 58, no. 10, pp. 2845–2854, 2010

    work page 2010

  7. [7]

    Skewed coding for suppression of pattern-dependent errors,

    A. Shafarenko, K. S. Turitsyn, and S. K. Turitsyn, “Skewed coding for suppression of pattern-dependent errors,” inProc. 31st Europ. Conf. Optical Commun. (ECOC), Stevenage, UK, 2005, pp. 193–194

    work page 2005

  8. [8]

    Shaping codes for structured data

    Y . Liu and P. H. Siegel, “Shaping codes for structured data”, in2016 IEEE Global Communications Conference (GLOBECOM), Washington DC, USA, 2016, pp. 1–6

    work page 2016

  9. [9]

    Improved Gilbert-Varshamov bound for constrained systems,

    B. H. Marcus and R. M. Roth, “Improved Gilbert-Varshamov bound for constrained systems,”IEEE Trans. Inform. Theory, vol. 38, no. 4, pp. 1213–1221, July 1992

    work page 1992

  10. [10]

    Design techniques for weakly constrained codes,

    M. Jin, K. A. S. Immink, and B. F. Boroujeny, “Design techniques for weakly constrained codes,”IEEE Trans. Commun., vol. 51, no. 5, pp. 709–714, 2003

    work page 2003

  11. [11]

    A DNA-based archival storage system,

    J. Bornholt, et al., “A DNA-based archival storage system,” inProc. Twenty-First Int. Conf. Architectural Aupport for Programming Lan- guages and Operating Systems (ASPLOS), Mar. 2016, pp. 637–649

    work page 2016

  12. [12]

    Characterizing and measuring bias in sequence data,

    M. G. Ross, et al. “Characterizing and measuring bias in sequence data,” Genome Biology, vol. 14, no. 5, pp. 1–20, 2013

    work page 2013

  13. [13]

    Concentration inequalities for Markov chains by Marton couplings and spectral methods,

    D. Paulin, “Concentration inequalities for Markov chains by Marton couplings and spectral methods,”Electron. J. Probab., vol. 20, no. 79, pp. 1–32, 2015

    work page 2015

  14. [14]

    Constrained systems and cod- ing for recording channels,

    B. Marcus, R. M. Roth, and P. H. Siegel, “Constrained systems and cod- ing for recording channels,” inHandbook of Coding Theory(R. Brualdi, C. Huffman, and V . Pless, eds.), Elsevier, 1998

    work page 1998

  15. [15]

    An introduction to coding for constrained systems,

    B. H. Marcus, R. M. Roth, and P. H. Siegel, “An introduction to coding for constrained systems,” lecture notes

    work page

  16. [16]

    Semiconstrained sys- tems,

    O. Elishco, T. Meyerovitch, and M. Schwartz, “Semiconstrained sys- tems,”IEEE Trans. Inf. Theory, vol. 62, no. 4, pp. 1688–1702, 2016

    work page 2016

  17. [17]

    Weakly constrained codes via row-by- row coding,

    S. Buzaglo and P. H. Siegel, “Weakly constrained codes via row-by- row coding,” inProc. 2017 IEEE Inform. Theory Workshop (ITW), Kaohsiung, Taiwan, 2017, pp. 151–155

    work page 2017

  18. [18]

    Error-resilient weakly constrained coding via row-by-row coding,

    P. Mishra and N. Kashyap, “Error-resilient weakly constrained coding via row-by-row coding,” inIEEE Int. Symp. Inf. Theory (ISIT), Athens, Greece, July 2024, pp. 1251–1256

    work page 2024

  19. [19]

    Dembo and O

    A. Dembo and O. Zeitouni,Large Deviations Techniques and Applica- tions, 2nd ed., Springer, Berlin, 2010

    work page 2010

  20. [20]

    D. A. Levin, and Y . Peres,Markov Chains and Mixing Times, 2nd ed., Amer. Math. Soc. Providence, RI, 2017

    work page 2017

  21. [21]

    Statistical methods in Markov chains,

    P. Billingsley, “Statistical methods in Markov chains,”Ann. Math. Statist., vol. 32, pp. 12–40, 1961

    work page 1961

  22. [22]

    Some distribution and moment formulae for the Markov chain,

    P. Whittle, “Some distribution and moment formulae for the Markov chain,”J. R. Stat. Soc. Ser. B Stat. Methodol., vol. 17, no. 2, pp. 235– 242, 1955

    work page 1955

  23. [23]

    On the zero- error capacity threshold for deletion channels,

    I. A. Kash, M. Mitzenmacher, J. Thaler, and J. Ullman, “On the zero- error capacity threshold for deletion channels,” inProc. 2011 Inf. Theory and Applications Workshop, La Jolla, CA, USA, pp. 1–5, 2011

    work page 2011