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arxiv: 2501.07561 · v5 · pith:OGUNPBKAnew · submitted 2025-01-13 · 💻 cs.IT · math.IT

Design and Analysis of a Concatenated Code for Intersymbol Interference Wiretap Channels

Aria Nouri , Reza Asvadi , Jun Chen This is my paper

Pith reviewed 2026-05-23 05:09 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords concatenated codingISI wiretap channelsLDPC codestrellis codessecrecy capacityinformation leakagedegree distribution optimization
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The pith

A concatenated LDPC-trellis scheme achieves tight lower bounds on secrecy capacity for ISI wiretap channels while driving the leakage upper bound to zero.

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

The paper first derives the secrecy capacity of intersymbol interference wiretap channels. It then constructs a practical two-stage code that places irregular LDPC codes in the outer layer and specially designed trellis codes in the inner layer. The trellis stage converts the uniform LDPC output into a Markov source whose statistics match those required by the capacity expression. Degree-distribution optimization on the LDPC stage is shown to force the estimated upper bound on information leakage toward zero, thereby satisfying the weak secrecy criterion without sacrificing the achievable rate.

Core claim

The central claim is that the proposed concatenated scheme, built from outer LDPC codes nested inside inner trellis codes, attains tight lower bounds on the secrecy capacity of ISI wiretap channels; further, optimization of the irregular LDPC degree distributions reduces the upper bound on the information leakage rate to zero, meeting the weak secrecy criterion.

What carries the argument

The nested wiretap-code structure in which outer LDPC codes generate uniformly distributed words that inner trellis codes reshape into a Markov process whose transition probabilities achieve the secrecy-capacity lower bound.

If this is right

  • Reliable rates arbitrarily close to the secrecy capacity become achievable with finite-length codes over ISI wiretap channels.
  • The same concatenated architecture can be reused for any channel whose secrecy capacity is achieved by a Markov input distribution.
  • Weak secrecy is obtained solely by degree optimization rather than by explicit randomness extraction at the encoder.

Where Pith is reading between the lines

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

  • If the inner trellis stage can be made rate-preserving for other memory channels, the same outer-code optimization may extend to non-ISI wiretap models.
  • The leakage-bound reduction to zero suggests that finite-length LDPC ensembles can be tuned to satisfy strong secrecy in the limit, though the paper stops at the weak criterion.

Load-bearing premise

A trellis code exists that maps any uniform LDPC codeword sequence into the exact Markov process required by the secrecy capacity without rate loss or extra leakage.

What would settle it

Numerical optimization of the LDPC degree distributions fails to drive the computed upper bound on leakage below a positive constant, or the designed trellis code produces a stationary distribution that deviates measurably from the capacity-achieving Markov chain.

Figures

Figures reproduced from arXiv: 2501.07561 by Aria Nouri, Jun Chen, Reza Asvadi.

Figure 1
Figure 1. Figure 1: (a) Trellis section of an FSMC, representing the EPR4 channel and the collection of branches βf (i, j) = (i, x, u, j) for all (i, j) ∈ Bˇf . (b) Trellis section of a 3-rd order E-FSMC, representing the DICODE channel and the collection of branches βe(i, ℓ, j) ≜ (i, x 3 , u 3 , j) for all (i, ℓ, j) ∈ Bˇe. between sf (t−1), x(t)  and sf (t−1), bf (t)  , for all t ∈ Z. In this case, the factor on the RHS of… view at source ↗
Figure 2
Figure 2. Figure 2: Trellis section of a 2-nd order E-FSMC that represents an ISI-WTC, comprising a DICODE channel as Bob’s channel and an EPR4 channel as Eve’s channel. Each branch of the trellis section is labeled by βe(i, ℓ, j) ≜ (i, x 2 , u 2 , v 2 , j) for all (i, ℓ, j) ∈ Bˇe. For any positive integer N, the joint PMF/PDF of B N e and Y nN conditioned on Se(0) = se(0) and X nN = x nN is pBN e ,YnN |Se (0),XnN b N e , y n… view at source ↗
Figure 3
Figure 3. Figure 3: Wolf-Ungerboeck code of rate 1/2 and the collection of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Normal factor graph representation of the superchannel. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trellis section of a superchannel resulted from concatenating the trellis code depicted in Fig. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Soft-output messages from FNs to VNs at Bob’s decoder. (Refer to equations ( [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Normal graph of the two-stage concatenated code, comprising a superchannel ( [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The i.u.d. secure rate R (4) iud, the constrained secrecy capacity R (4) csc of the 4-th order E-FSMC modeling the ISI-WTC in Example 1, and the i.u.d. secure rate of the superchannel R SC iud in Table V (Example 4) for SNRE dB = −4.0. Note that the superchannel is specifically designed for operating at SNRB dB =SNRE dB =−4.0. ordinal optimization. For the SNR values of Example 1, the i.u.d. secure rate of… view at source ↗
Figure 10
Figure 10. Figure 10: Multi-stage decoder for estimating m (l) , assuming that the symbols (m (1) , . . . , m (l−1)) are successfully decoded without error (1 ≤ l ≤ m). the asymptotic error probability of Frank’s decoder (ϵ F ≜ limN′→∞ ϵ F nN′ ) is approximated by applying the following modifications to the density evolution algorithm. (i) Since Frank’s and Eve’s observations are identical (Remark 6), to evolve f (ı) vf (ξ) th… view at source ↗
Figure 11
Figure 11. Figure 11: The average asymptotic performance of Bob’s, Eve’s, and [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (a): Gain-to-noise power spectrum ratios of Bob’s and Eve’s point-to-point channels in dB/Hz. (b): Power spectra of sequences produced by Markov sources optimized for the 1-st order and 4-th order E-FSMCs representing the considered ISI-WTC, and the power spectrum of a sequence generated by the Markov source specified in Table I, satisfying the upper bound constraint in (11). VI. FINITE BLOCKLENGTH REGIME… view at source ↗
Figure 13
Figure 13. Figure 13: Power spectra of sample codewords generated by the inner [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The bit error rate of the constructed code [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Time delays between transmitted signals from the antenna [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Power-delay profile of the multipath channel associated with [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Generalization of the one-time stochastic encoder in [ [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
read the original abstract

We propose a two-stage concatenated coding scheme for reliable and secure communication over intersymbol interference wiretap channels. We first establish the secrecy capacity. Then, motivated by the theoretical codes that achieve the secrecy capacity, our scheme integrates low-density parity-check (LDPC) codes in the outer stage, forming a nested structure of wiretap codes, with trellis codes in the inner stage to improve achievable secure rates. The trellis code is specifically designed to transform the uniformly distributed codewords produced by the LDPC code stage into a Markov process, achieving tight lower bounds on the secrecy capacity. We further estimate the information leakage rate of the proposed scheme using an upper bound. To meet the weak secrecy criterion, we optimize degree distributions of the irregular LDPC codes at the outer stage, essentially driving the estimated upper bound on the information leakage rate to zero.

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 proposes a two-stage concatenated coding scheme for ISI wiretap channels: an outer nested irregular LDPC wiretap code followed by an inner trellis code that maps uniform LDPC outputs to a stationary Markov process. It first claims to establish the secrecy capacity, then asserts that the scheme achieves tight lower bounds on this capacity; degree-distribution optimization is used to drive an upper bound on the information leakage rate to zero, meeting the weak secrecy criterion.

Significance. If the trellis mapping preserves the exact input statistics required for the secrecy-capacity lower bound without rate loss and if the leakage bound is independent of the optimized degrees, the construction would supply a practical, optimizable coding scheme for a non-trivial class of wiretap channels with memory. The explicit use of nested wiretap codes and the Markov-motivated inner stage are concrete contributions that could be useful for further work on channels with ISI.

major comments (2)
  1. [Abstract / trellis inner code] Abstract and trellis-code design section: the central claim that the inner trellis code converts uniform i.i.d. LDPC codewords into the precise stationary Markov process that attains the secrecy-capacity lower bound, without rate loss or secrecy degradation, is load-bearing; no generator matrix, state diagram, transition-probability verification, or rate calculation is referenced to confirm that the marginal distribution and rate are exactly preserved.
  2. [Leakage bound and degree optimization] Leakage estimation and LDPC optimization: the information leakage rate is bounded above and the bound is then driven to zero by optimizing the free parameters (LDPC degree distributions); it is unclear whether the upper bound expression remains independent of these parameters or whether the optimization step effectively defines the reported leakage value, which would undermine the claim of an independently verified achievable rate.
minor comments (2)
  1. [Abstract] The abstract states that secrecy capacity is established but supplies no outline of the derivation or the channel model assumptions; a short paragraph in the introduction would improve readability.
  2. [Notation] Notation for the Markov process statistics and the resulting secrecy rate expressions should be introduced once and used consistently; several symbols appear without prior definition in the abstract-level description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. Below we respond point-by-point to the major comments, indicating where revisions will be made to improve clarity.

read point-by-point responses

  1. Referee: [Abstract / trellis inner code] Abstract and trellis-code design section: the central claim that the inner trellis code converts uniform i.i.d. LDPC codewords into the precise stationary Markov process that attains the secrecy-capacity lower bound, without rate loss or secrecy degradation, is load-bearing; no generator matrix, state diagram, transition-probability verification, or rate calculation is referenced to confirm that the marginal distribution and rate are exactly preserved.

    Authors: The trellis code is constructed specifically to map the uniform i.i.d. outputs of the outer LDPC stage onto the exact stationary Markov process that attains the secrecy-capacity lower bound. We will revise the manuscript to include the generator matrix, state diagram, explicit transition-probability verification, and rate calculation demonstrating that the marginal distribution and rate are preserved without loss or secrecy degradation. revision: yes

  2. Referee: [Leakage bound and degree optimization] Leakage estimation and LDPC optimization: the information leakage rate is bounded above and the bound is then driven to zero by optimizing the free parameters (LDPC degree distributions); it is unclear whether the upper bound expression remains independent of these parameters or whether the optimization step effectively defines the reported leakage value, which would undermine the claim of an independently verified achievable rate.

    Authors: The upper bound on the information leakage rate is derived as an explicit function of the LDPC degree distributions (and other fixed parameters) prior to any optimization. The subsequent optimization identifies degree distributions for which the bound evaluates to zero, confirming that the weak secrecy criterion is met. The bound expression itself is independent of the particular optimized values; the optimization merely demonstrates that suitable parameters exist. We will add a clarifying sentence in the revised manuscript to emphasize this separation. revision: partial

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper first establishes the secrecy capacity of the ISI wiretap channel. It then constructs a concatenated scheme motivated by theoretical capacity-achieving codes, with the inner trellis code designed to map uniform LDPC outputs to the required Markov process and the outer irregular LDPC degree distributions optimized to drive an independent upper bound on leakage to zero. This is standard parameter optimization for meeting a design criterion rather than a fitted input renamed as prediction or any self-definitional reduction. No load-bearing self-citation, uniqueness theorem, or ansatz smuggling is present in the text. The central claims rest on explicit design choices whose correctness can be checked externally via the resulting rates and bounds.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The construction rests on standard information-theoretic assumptions about secrecy capacity existence and the ability to design a trellis code that matches channel memory without rate penalty; one free parameter is the LDPC degree distribution optimized to the leakage bound.

free parameters (1)
  • LDPC degree distributions
    Optimized at the outer stage to drive the estimated upper bound on information leakage rate to zero.
axioms (1)
  • domain assumption Secrecy capacity of the ISI wiretap channel can be established and used as a benchmark for the concatenated scheme.
    Invoked as the first step before constructing the practical code.

pith-pipeline@v0.9.0 · 5674 in / 1292 out tokens · 68945 ms · 2026-05-23T05:09:28.832385+00:00 · methodology

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