On the Synthetic Channels in Polar Codes over Binary-Input Discrete Memoryless Channels

Ming Xu; Xiaoyan Cheng; Yadong Jiao; Yuansheng Tang

REVIEW 1 major objections 5 minor 20 references

On the Synthetic Channels in Polar Codes over Binary-Input Discrete Memoryless Channels

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that every synthetic channel arising in polar codes over symmetric binary-input discrete memoryless channels has a compact representation as a random switching of few binary symmetric channels, with an explicit polynomial…

desk verdict Solid algebraic core, novel closed forms for synthetic channels, but Theorem 10's proof misses the B(0) edge case it claims to cover. read the letter →

arxiv 2506.04163 v1 pith:OVHCEWNV submitted 2025-06-04 cs.IT math.IT

Yadong Jiao , Xiaoyan Cheng , Yuansheng Tang , Ming Xu This is my paper

classification cs.ITmath.IT MSC 94A15

keywords polarcodessyntheticchannelsBIDMClikelihoodratioprofilerandomswitchingchannelbinarysymmetricpolarizationreliabilityevaluation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the synthetic channels produced while constructing polar codes over symmetric binary-input discrete memoryless channels have compact algebraic representations. It defines a channel's likelihood ratio profile, uses it to define an equivalence of channels, and shows that every symmetric channel is equivalent to a random switching of binary symmetric channels. The main result bounds the number of binary-symmetric components needed for each synthetic channel and gives a lower bound on how many output symbols share the same likelihood ratio. If the claim holds, the reliability of these synthetic channels can be computed efficiently, not only for erasure channels but for all symmetric channels.

What carries the argument

The load-bearing objects are the likelihood ratio profile $P_W(\varepsilon)$, the probability that a uniformly random input produces an output with likelihood ratio $\varepsilon$, and the random switching decomposition $W=\sum_j q_j W_j$, where the sub-channel used is known to the receiver. A symmetric channel has a profile symmetric about $1/2$ and is therefore equivalent to an RSC of BSCs $B(\varepsilon)$. The paper defines two binary operations on crossover probabilities, $\varepsilon_0\star\varepsilon_1=\bar{\varepsilon}_0\varepsilon_1+\varepsilon_0\bar{\varepsilon}_1$ and $\varepsilon_0\diamond\varepsilon_1=\varepsilon_0\varepsilon_1/(\varepsilon_0\star\varepsilon_1)$ (with endpoint conventions), which mirror the two polar transforms: the 'sum' transform of two BSCs is a BSC with crossover $\varepsilon_0\star\varepsilon_1$, while the 'product' transform splits into a two-component RSC with crossovers $\varepsilon_0\diamond\varepsilon_1$ and its complement. These operations allow the recursive generation of synthetic channels to be unfolded into explicit sums of BSCs, and the counting argument bounds how many distinct components remain after merging.

What would settle it

Take $W$ to be a nontrivial mixture of a noiseless binary symmetric channel and a positive-crossover binary symmetric channel, and compute the exact LRP-oriented RSC form of $A_{01}(W)$ ($k=2$, $\alpha=01$) for small $n$. If its number of distinct BSC components exceeds $h_{01}(n)$, or if the average number of output symbols per likelihood-ratio level is below $\varphi(01)/2$ for large $n$, then the theorem's extension to $\varepsilon_0=0$ fails.

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Extended reading notes

Core claim

The paper's central claim is Theorem 10: for any symmetric BIDMC $W$ equivalent to a random switching of BSCs with crossover probabilities $0\le\varepsilon_0<\cdots<\varepsilon_{n-1}<\varepsilon_n=1/2$, and any $\alpha\in\{0,1\}^k$, the synthetic channel $A_\alpha(W)$ is equivalent to a random switching of at most $h_\alpha(n)=2^{b(\alpha)}(2n)^{2^k}/\varphi(\alpha)+g_\alpha(n)$ BSCs, where $g_\alpha(n)$ is a polynomial of degree at most $2^k-1$ and $\varphi(\alpha)$ is an explicitly defined integer. Consequently, for sufficiently large $n$, the average number of output symbols sharing a single likelihood ratio is at least $\varphi(\alpha)/2$. The proof converts the two basic polar transforms into algebraic operations on crossover probabilities and then counts the number of distinct BSC components that can survive after merging equivalent components; exact LRP-oriented forms are also given for several families of synthetic channels when the underlying channel is a BSC.

Load-bearing premise

The proof of the main bound applies a component-count estimate that was established only for random switching channels whose binary symmetric components all have strictly positive crossover probability, but the induction can create a noiseless component; if the estimate fails there, both the polynomial bound and the multiplicity lower bound are unsupported.

Editorial extensions

If this is right

For a fixed polar-code order $k$, the number of BSC components in the compact representation of any synthetic channel grows polynomially in $n$, so the representation remains small relative to the exponentially large output alphabet.
Capacity, maximum-likelihood error probability, and Bhattacharyya parameter of each synthetic channel can be computed from the compact RSC form using the profile formulas of Theorem 1, without enumerating the output alphabet.
The lower bound $\varphi(\alpha)/2$ on the average multiplicity of likelihood-ratio values shows that the refined output alphabet cannot fragment indefinitely as $n$ grows.
For an underlying BSC, the exact LRP-oriented forms in Theorem 9 give closed-form descriptions of several families of synthetic channels, usable directly in code construction.
The results extend exact reliability evaluation beyond the binary erasure channel, where recursive evaluation was previously the only tractable case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step, not taken in the paper, is to turn the polynomial bound into an explicit algorithm that computes the LRP-oriented RSC form of $A_\alpha(W)$ in time polynomial in $n$ for fixed $k$; the paper stops at showing the representation exists and is compact.
The same likelihood-ratio-profile framework could be applied to asymmetric BIDMCs by tracking the two one-sided profiles or by symmetrizing first, though the algebraic simplification to $\star$ and $\diamond$ relies on symmetry.
The lower bound on average multiplicity may be only a leading-order estimate; exact counts for small $k$ and $n$ would show how tight the constant $\varphi(\alpha)/2$ is.
The compact representation suggests a quantitative complement to partial-order methods: once a partial order identifies which synthetic channels are worst, the RSC form could be used to evaluate only those channels exactly.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Name],

You should know: this paper gives the first closed-form expression for the number of distinct likelihood-ratio classes in the synthetic channels of polar codes over symmetric BIDMCs, as an explicit polynomial in the output-alphabet size. The main theorem (Theorem 10) states that for any α ∈ {0,1}^k, A_α(W) is equivalent to an RSC of at most h_α(n) = 2^{b(α)}(2n)^{2^k}/φ(α) + O(n^{2^k−1}) BSCs, so the average output multiplicity per likelihood ratio is at least φ(α)/2 for large n. That result is new, as far as I can tell from the cited literature.

What's genuinely good: the likelihood ratio profile (LRP) framework is clean. Theorems 1–6 are proved carefully; the equivalence and degradation results are correct. The algebraic decomposition of symmetric BIDMCs into BSCs and the star/diamond operations for one Arikan step are standard, but the iterative closed forms in Theorems 7 and 8 go beyond what I've seen in the references. Theorem 9's explicit LRP-oriented forms for specific bit sequences are dense but credible; I couldn't check every coefficient, but the logic is consistent. The paper is self-contained, definitions are precise, and there is no fitting or circularity.

The soft spot: the proof of Theorem 10 does not cover the case ε_0 = 0, even though the theorem statement allows it. Theorem 8, on which the proof relies, is proved only for 0 < ε_0 < ... < ε_n = 1/2. When W has a B(0) component (e.g., a BEC), the synthetic channels can also contain B(0); A_0(B(0),·) acts as identity and A_1(B(0),·) is absorbing. The induction step and the base cases both inherit this gap. This is load-bearing because the final average-multiplicity bound depends on it. The gap is likely repairable—Lemma 8(3) suggests a B(0) term contributes at most a lower-order polynomial—but it has to be written out. As printed, the theorem statement overclaims.

Also, the abstract overstates: it says there is 'no effective approach' to evaluate reliability except for BECs, which ignores density evolution, Tal–Vardy upgrading/degrading, and the partial-order methods the paper itself cites. The qualifiers 'symmetric' and 'for sufficiently large n' are missing from the abstract, which is misleading.

Who it's for: researchers working on exact analysis or construction of polar codes from symmetric DMCs. It deserves a serious referee: the core is sound, the new closed forms are useful, and the ε_0 = 0 gap is a fixable but essential repair. I'd send it to review with a request to extend Theorem 8 (or restrict Theorem 10) and to rewrite the abstract.

My recommendation: conditional accept after the edge case is handled.

Referee Report

1 major / 5 minor

Summary. The paper develops an algebraic framework for studying the synthetic channels that arise in Arikan polar-code construction over symmetric binary-input discrete memoryless channels (BIDMCs). It defines equivalence of BIDMCs through the likelihood ratio profile (LRP), introduces a notion of symmetry based on the LRP, and shows that symmetric BIDMCs are equivalent to random switching channels (RSCs) of binary symmetric channels (BSCs). The central technical results are explicit algebraic descriptions of the Arikan transformations A0 and A1: Theorem 5 decomposes transformations of RSCs, Lemmas 4 and 6 give formulas for transformations of BSCs, and Theorems 7 and 8 bound the number of BSCs needed to represent the iterated transformations Δ_m(W) and ∇_m(W). Theorem 9 gives LRP-oriented forms for several synthetic channels when the underlying channel is a BSC. The main quantitative claim is Theorem 10: for W ∼= Σ_{j∈[n+1]} q_j B(ε_j) with 0 ≤ ε_0 < ... < ε_n = 1/2, the number of LRP classes in A_α(W) is at most h_α(n) = 2^{b(α)}(2n)^{2^k}/φ(α) + g_α(n) with g_α polynomial of degree at most 2^k−1, and consequently the average number of output symbols sharing one likelihood ratio is at least φ(α)/2 for sufficiently large n.

Significance. If the proof is completed, the paper provides a genuinely new quantitative handle on polar-code synthetic channels: it replaces an exponentially large output alphabet by an RSC representation whose size is polynomial in the number of BSCs in the underlying channel, for fixed code order k. The paper is self-contained, and Theorems 1 through 6 are proved cleanly; Theorems 7 through 9 provide explicit, checkable formulas. Theorem 10 is a falsifiable asymptotic statement and, if established in the stated generality, would be a useful contribution to the understanding of reliability evaluation for symmetric BIDMCs. The main reservation is the ε_0 = 0 edge case: as printed, the proof of Theorem 10 does not cover channels containing a noiseless B(0) component, even though the theorem statement explicitly allows ε_0 = 0. This gap is localized and likely repairable, but it is load-bearing for the final average-multiplicity claim.

major comments (1)

[Section 4.3, Theorem 10] The statement of Theorem 10 allows 0 ≤ ε_0 < ... < ε_n = 1/2, but the proof applies Theorem 8 in the base cases and in the induction step. Theorem 8 is stated and proved only for 0 < ε_0 < ... < ε_n = 1/2. This is not a cosmetic mismatch: Lemma 5 shows A1(B(0),W) ∼= B(0) and A0(B(0),W) ∼= W, so if the underlying channel has a B(0) component, the synthetic channels A_α(W) can themselves contain B(0) components. The identities and absorbing behavior of B(0) are not accounted for in the component-count bound of Theorem 8. Consequently, the bound h_α(n) in Eq. (90), and hence the final 'at least φ(α)/2' average-multiplicity lower bound, are not rigorously established for the ε_0 = 0 case included in the theorem statement. The gap is likely repairable through Lemma 8(3), but the repair must be written out before the result is complete.

minor comments (5)

[Section 2.2, last paragraph] The sentence 'for any p∈(0,1) with p≠1/2 the channel given in Figure 2 is symmetric according to our definition, but according to that noted in [3]' is incomplete; it should end with something like 'but it is not symmetric in the sense of [3].'
[Theorem 8, first display] The equality 'H_{m,n−1}+1 = C(m+n−1,m)+1' is inconsistent with Lemma 9's definition H_{m,n} = C(m+n−1,m); with n non-1/2 BSCs the bound should read H_{m,n}+1.
[Lemma 2, proof] The Jensen step is applied to the concave functions ℏ(ε), min{ε, ε̄}, and √(εε̄); calling them convex is a terminology slip, though the displayed inequalities are the correct ones.
[Lemma 1, proof] The ratio ε′_0/ε̄′_0 is not defined when ε′_0 equals 0 or 1; the support inequalities are still true but require separate trivial arguments for these endpoints.
[Abstract] The phrase 'the synthetic channels, which and their compounds are called Arikan transformations' is grammatically awkward and should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a self-contained algebraic derivation of compact RSC representations and component-count bounds for polar synthetic channels.

full rationale

The paper is a fully self-contained mathematical derivation. The likelihood ratio profile is defined directly from the channel transition probabilities (eqs. 4-5), and Theorem 1 derives I(W), P_e(W), and Z(W) from that profile by direct algebraic manipulation. The RSC formalism is introduced by definition (eq. 11), and Theorems 2, 5, Lemma 4, Lemma 6, and Theorems 6-9 derive the synthetic-channel representations from the BSC transition laws and the Arikan transformation definitions (eqs. 21-22). No parameter is fitted to data and later renamed a prediction; no conclusion is assumed among the premises; and no load-bearing claim rests on a citation to the authors' own prior work. The central bound in Theorem 10 is obtained by induction from the explicit component-count bounds of Theorem 8, and the final multiplicity lower bound follows arithmetically from the size of the output alphabet, not from the statement being proved. The reviewer-noted gap that Theorem 10 allows epsilon_0 = 0 while Theorem 8 is stated only for 0 < epsilon_0 is a correctness/rigor concern about extending the bound to channels with a noiseless component; it is not circularity, because the extension is not achieved by assuming the theorem's conclusion. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation is closed-form: no parameters are fit and no new physical entities are introduced. The central claim rests on standard sufficient-statistic facts, the uniform-input BIDMC model, the Arikan transform definition, and one implicit edge-case extension of the bound to noiseless components.

assumptions (4)

standard math The likelihood ratio is a sufficient statistic for decoding a BIDMC, so channels with the same LRP have identical I, P_e, and Z (Theorem 1).
Proved in the paper via Theorem 1; this is a standard result, also cited to Richardson-Urbanke [2].
domain assumption The input x is uniformly distributed over F_2 and the output alphabet is discrete.
Stated at the start of Section 2; all definitions of capacity, LRP, and RSC rely on uniform inputs and discrete outputs.
standard math Arikan's 2x2 kernel transformation defines the synthetic channels A0 and A1, and polar codes are formed by iterating them (Lemma 7 from [3]).
The paper adopts the standard polar code construction; this is prior literature, not derived here.
ad hoc to paper The component-count bound of Theorem 8 remains valid when the channel decomposition includes a noiseless B(0) component; Theorem 10 uses this in its induction step.
Theorem 8 is stated for 0 < ε_0, but Theorem 10's setup allows ε_0 = 0 and its proof applies Theorem 8 to such channels without comment.

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Cite this review

Pith. "Pith review of On the Synthetic Channels in Polar Codes over Binary-Input Discrete Memoryless Channels." pith.science (2026). https://pith.science/paper/OVHCEWNV

@misc{pith2026250604163,
  author       = {Pith},
  title        = {Pith review of: On the Synthetic Channels in Polar Codes over Binary-Input Discrete Memoryless Channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVHCEWNV}},
  note         = {Machine review of arXiv:2506.04163}
}

read the original abstract

Polar codes introduced by Arikan in 2009 are the first code family achieving the capacity of binary-input discrete memoryless channels (BIDMCs) with low-complexity encoding and decoding. Identifying unreliable synthetic channels in polar code construction is crucial. Currently, because of the large size of the output alphabets of synthetic channels, there is no effective approach to evaluate their reliability, except in the case that the underlying channels are binary erasure channels. This paper defines equivalence and symmetry based on the likelihood ratio profile of BIDMCs and characterizes symmetric BIDMCs as random switching channels (RSCs) of binary symmetric channels. By converting the generation of synthetic channels in polar code construction into algebraic operations on underlying channels, some compact representations of RSCs for these synthetic channels are derived. Moreover, a lower bound for the average number of elements that possess the same likelihood ratio within the output alphabet of any synthetic channel generated in polar codes is also derived.

Figures

Figures reproduced from arXiv: 2506.04163 by the authors.

**Figure 1.** An RSC of BIDMCs {Wj}j∈[n] . For each transmission, the subchannel Wj is chosen with probability qj to be the actually used channel. is independent of a ∈ X . Clearly, we have qj = Pr(y ∈ Yj ), j ∈ [n] and that (q0, q1, . . . , qn−1) is a probability distribution vector. For j ∈ [n], let Wj : x ∈ X 7→ yj ∈ Yj denote the synthetic BIDMC with transition probabilities Pr(yj |x) = 1 qj Pr(y = yj |x), (10) which, in acc… view at source ↗

**Figure 2.** A symmetric BIDMC which is equivalent to B( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Synthetic channels in polar code of order [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Extremes of information combining,

I. Sutskover, S. Shamai (Shitz) and J. Ziv, “Extremes of information combining,”IEEE Trans. Inf. Theory,vol. 51, no. 4, pp. 1313-1325, Apr. 2005

work page 2005

[2] [2]

Richardson and R

T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge, U.K.: Cambridge Univ. Press, 2008

2008

[3] [3]

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, Jul. 2009

2009

[4] [4]

Performance of polar codes with the construc- tion using density evolution,

R. Mori and T. Tanaka, “Performance of polar codes with the construc- tion using density evolution,”IEEE Commun. Lett.,vol. 13, no. 7, pp. 519-521, Jul. 2009

work page 2009

[5] [5]

Polar codes for channel and source coding,

S. B. Korada, “Polar codes for channel and source coding,” Ph.D. disser- tation, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzer- land, 2009

work page 2009

[6] [6]

Efficient design and decoding of polar codes,

P. Trifonov, “Efficient design and decoding of polar codes,”IEEE Trans. Commun.,vol. 60, no. 11, pp. 3221-3227, Nov. 2012

work page 2012

[7] [7]

How to construct polar codes,

I. Tal and A. Vardy, “How to construct polar codes,”IEEE Trans. Inf. Theory,vol. 59, no. 10, pp. 6562-6582, Oct. 2013

2013

[8] [8]

A partial order for the synthesized channels of a polar code,

C. Schurch, “A partial order for the synthesized channels of a polar code,”in Proc. IEEE Int. Symp. Inf. Theory (ISIT),Barcelona, Spain, pp. 220-224, Jul. 2016

work page 2016