# Coset Codes—Part I: Introduction and Geometrical Classification

**Date**: 03/09/2026
**Domain**: computer_science/information_theory
**Taxonomy**: academic/research_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Classification of Eight Code Classes Is Enumerative Rather Than Principled**

Section VI introduces eight trellis code classes defined by three binary characteristics (partition type, encoder rate, constraint length), suggesting a clean 2×2×2 taxonomy. However, the paper itself acknowledges that Class I and Class V coincide for two-state codes (k=1), that Class II and Class VI coincide for four-state D₄/RD₄ codes, and that the first Class VII and Class III codes are identical. These overlaps indicate that the three binary characteristics do not generate eight genuinely distinct classes across all parameter values. Additionally, the boundary between CS-type and Wei-type codes is defined informally in Section V rather than derived from the Section VI classification criteria, and the paper does not establish that the three chosen dimensions span the full relevant design space. It would be helpful to either provide a formal criterion that partitions the code space without overlap—for example, based on trellis state-space dimension relative to partition order—or to explicitly acknowledge that the eight classes are representative exemplars rather than a complete, mutually exclusive partition of the design space.

**Fundamental Coding Gain Conflates Asymptotic and Finite-SNR Regimes Without Adequate Separation**

The paper defines γ(C) = 2^{−ρ(C)} d²_min(C) as the primary performance metric and uses it throughout for cross-class comparisons, while simultaneously introducing an effective coding gain γ_eff via the rule of thumb that every factor-of-two increase in error coefficient Ñ₀ costs approximately 0.2 dB. This creates a structural inconsistency: γ(C) is an asymptotic quantity valid only at high SNR and for large constellations, yet γ_eff is a finite-SNR correction applied on top of it. The 0.2 dB rule is applied uniformly across codes whose error coefficients range from 4 to over 9,000 (Table XI), a range over which the approximation quality likely varies substantially, and γ_eff estimates are absent for several Class V–VIII entries where error coefficients are unknown or very large. The folk theorem in Section VII then draws comparative conclusions based on a mixture of γ and γ_eff values without flagging which data points lack reliable γ_eff estimates. Readers might note that for codes with very large Ñ₀, the fundamental coding gain figure is essentially uninformative as a predictor of actual performance. It would be helpful to bound the error in the 0.2 dB approximation, to flag explicitly which Table XI entries lack reliable γ_eff estimates, and to qualify the folk theorem accordingly.

**Partition Regularity and Labeling Canonicality Are Asserted but Not Verified for Higher-Depth Partitions**

Section IV-C introduces the notion of a regular labeling and asserts that regular labelings exist for all partitions used in the paper, but verification is provided only for three specific sufficient conditions (linear labelings, Ungerboeck labelings where the distance bound holds with equality, and Cartesian product labelings). For mod-4 and higher-depth partitions—such as E₈/RE₈ (depth 3), Λ₁₆/RΛ₁₆ (depth 4), and the two-level chains used in Classes III–IV and VII–VIII—neither condition is explicitly verified, and the claim is deferred to Part II without a precise theorem reference. The Class III distance analysis invokes an 'alternative partition E₈/R*E₈/2E₈' to ensure the Ungerboeck bound holds at merge points, but the existence and properties of R*E₈ are asserted without derivation. Since distance-invariance—and hence the validity of d²_min calculations and error coefficient values in Table XI—depends entirely on regularity, any gap here propagates to the coding gain figures for several code classes. Furthermore, for higher-dimensional partitions such as E₈/RE₈ (order 16) or Λ₁₆/RΛ₁₆ (order 256), the paper does not address whether different choices of coset representatives yield inequivalent codes or merely isomorphic ones, so it is unclear whether reported error coefficients correspond to the best known labeling or an arbitrary one. It would be helpful to either supply regularity proofs inline for each partition appearing in Tables VII–XI, or to give precise theorem references to Part II for each case, and to clarify the labeling-dependence of the reported error coefficients.

**Dual Lattice Construction Relies on a Depth-Dependent Definition Not Shown to Be Fully Consistent**

Section II-H defines the dual lattice Λ⊥ using the complex inner product modulo φ^μ and notes that this definition coincides with the standard real dual only when the φ-depth μ is even; for odd μ, the two definitions yield lattices differing by a rotation R. This discrepancy is acknowledged but not resolved: the paper asserts that the complex definition is 'more natural' without verifying that the partition properties and distance parameters attributed to dual lattices in Table III—and used in the coding gain analysis of Sections V–VI—are consistent under both conventions. In particular, the self-duality of D₄ and E₈ is verified only under the complex definition, while real code formulas in Table I use the real definition. Additionally, the proof sketch for the dual code formula (reversing the chain (C₀,…,C_{μ-1}) to (C₀⊥,…,C_{μ-1}⊥)) relies on generators being binary-valued, which holds for decomposable lattices but is not argued for the indecomposable cases (Λ₂₄, H₂₄) listed in Table I with prime notation. It would be helpful to add a lemma confirming that all partition properties cited in Table III are invariant to the choice of duality convention, or to adopt a single consistent definition throughout and handle the indecomposable cases separately.

**Decomposability Assumptions for Depth μ ≥ 3 and Indecomposable Lattices Are Incompletely Verified**

Section II-G establishes that decomposable complex binary G-lattices of depth μ have code formulas of the form Λ = φ^μ G^N + φ^{μ-1} C_{μ-1} + … + C₀, and the minimum squared distance formula and dual lattice construction of Section II-H both rely on this decomposability. However, the proof that Λ is a G-lattice (property a) is given fully only for the mod-2 case in Lemma 4; for depth μ ≥ 3, the result is stated without a complete proof, with the paper noting only that 'the ones we are concerned with generally are' decomposable. The Leech lattice Λ₂₄ and H₂₄ are explicitly flagged as indecomposable, yet their code formulas appear in Table I with a prime notation defined only informally, and the self-duality of Λ₂₄ invoked in Section III to motivate its inclusion in Table II rests on the dual construction whose extension to indecomposable lattices is unverified. Since coding gain values for partitions involving Λ₂₄-related lattices in Tables II–III depend on these structural claims, it would be helpful to either provide the missing proofs for depth μ ≥ 3 or to clearly delineate which results in Tables II and III depend on the indecomposable case and require separate justification in Part II.

**The 'Folk Theorem' on States-vs.-Gain Trade-off Is Presented Without Sufficient Qualification**

Section VII proposes a folk theorem asserting specific encoder state counts required to achieve specific coding gains (2 states → 1.5 dB, 4 states → 3 dB, 16 states → 4.5 dB, ~64 states → 5.25 dB, 256 states → 6 dB). However, the paper's own data in Table XI shows that Class V codes based on H₁₆/Λ₁₆ achieve γ = 2^{9/4} (≈6.77 dB) with only 64 states, and Class I codes based on Λ₁₆/RΛ₁₆ achieve 6 dB with 16 states—both apparently violating the folk theorem's implied monotonicity. These cases are dismissed due to large error coefficients, but no quantitative threshold for an 'acceptable' Ñ₀ is provided, leaving the theorem's domain of validity undefined. Furthermore, the theorem conflates encoder states 2^ν with effective decoding complexity, which for high-dimensional codes is dominated by the partition decoder rather than the trellis. The folk theorem is drawn from a specific set of codes sharing particular lattice families and redundancy constraints, and it is not compared against known information-theoretic lower bounds on decoding complexity. It would be helpful to restate the folk theorem as a conditional result with an explicit bound on Ñ₀, to clarify whether 'states' refers to encoder states or effective partition complexity, and to note explicitly that it is an empirical observation from the surveyed codes rather than a proven bound.

**Minimum-distance sufficiency assumption may not account for the full distance spectrum in BER prediction**

The paper's central performance metric is the fundamental coding gain γ(C) = 2^{−ρ(C)} · d²_min(C), which depends solely on the minimum squared Euclidean distance and the normalized redundancy. The effective coding gain γ_eff is then adjusted by a rule of thumb: 'every factor of two increase in the error coefficient reduces the coding gain by about 0.2 dB (at error rates of the order of 10⁻⁶).' This rule accounts for Ñ₀ (nearest neighbors) and, for Ungerboeck-type codes, Ñ₁ and Ñ₂ as well. However, the paper explicitly acknowledges that 'in principle, the error coefficients at every distance ought to be considered, and the effective coding gain evaluated in the same way for each; if they grow too large too rapidly, they can dominate performance.' For several code families—notably the Class V and VI codes with very large Ñ₀ (e.g., the 128-state Class V Λ₁₆/RΛ₁₆ code and the 16-state Class VI Λ₁₆/RΛ₁₆ code with Ñ_D ~ 2^19)—the paper omits γ_eff entirely or notes the error coefficient is 'huge.' Readers might note that for these high-gain, high-error-coefficient codes, the single-distance approximation may be a poor proxy for actual BER at operationally relevant error rates, and simulation validation or a full union-bound truncation analysis would strengthen the comparison. It would be helpful to clarify the range of SNR and error-rate regimes over which the 0.2 dB/factor-of-two rule of thumb remains a reliable approximation, particularly for codes where Ñ₀ exceeds several hundred.

**Assumption of distance-invariance requires verification for non-linear trellis codes built on mod-4 partitions**

The paper defines distance-invariance (Section IV-B) as the property that 'the distribution of distances from any given code sequence to all other code sequences is the same as the norm distribution of code sequences,' and states that 'all codes in this paper are distance-invariant.' Distance-invariance is established rigorously for linear trellis codes (Lemma 6, mod-2 partitions) and for codes with regular labelings. For mod-4 and deeper partitions, the paper asserts regularity under three sufficient conditions (linearity, Ungerboeck labeling with equality in the distance bound, or Cartesian-product structure), and states that 'regular labelings (although not necessarily regular Ungerboeck labelings) exist for all partitions used in all the codes covered in this paper.' However, the paper does not provide an explicit verification for each of the non-linear codes in Tables VII–XI that use mod-4 or mod-3 partitions (e.g., codes based on E₈/RE₈ or Λ₁₆/RΛ₁₆). It is not entirely clear how regularity is established for the indecomposable cases (e.g., Λ₂₄ and H₂₄, noted as indecomposable). If distance-invariance fails for any of these codes, the reported Ñ₀ values—computed as the norm distribution of sequences from the all-zero sequence—would not correctly represent the average number of nearest-neighbor error events, which would affect the validity of γ_eff. It would be helpful to either cite the specific regularity argument for each code family or note which codes rely on numerical verification rather than the general lemmas.

**Status**: [Pending]

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## Detailed Comments (25)

### 1. Volume Chain Contains Spurious Middle Term

**Status**: [Pending]

**Quote**:
> Since $V(\mathbb{C}) = V(\Lambda_0) = 2^{-k(C)}V(\Lambda') = 2^{k+r}V(\Lambda') = 2^{r(C)}V(\Lambda)$

**Feedback**:
The chain of equalities contains an inconsistent middle term. From Lemma 1, V(Λ₀) = 2^{−k} V(Λ'), and V(Λ') = 2^{k+r} V(Λ), giving V(Λ₀) = 2^r V(Λ). The term 2^{k+r}V(Λ') equals V(Λ') scaled up by 2^{k+r}, which is not equal to V(Λ₀) = 2^{−k}V(Λ'). The correct chain is V(C) = V(Λ₀) = 2^{−k(C)}V(Λ') = 2^{r(C)}V(Λ), dropping the spurious middle term 2^{k+r}V(Λ').

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### 2. Coding Gain Formula References Sublattice Distance Instead of Lattice Distance

**Status**: [Pending]

**Quote**:
> Relative to $\gamma (\Lambda) = 2^{-\rho (\Lambda)}d_{\min}^{2}(\Lambda^{\prime})$, the gain $\gamma (\mathbb{C})$ is greater by the distance gain factor of $d_{\min}^{2}(\mathbb{C}) / d_{\min}^{2}(\Lambda)$

**Feedback**:
The standard definition is γ(Λ) = 2^{−ρ(Λ)} d²_min(Λ), using the minimum distance of Λ itself. The expression as written substitutes d²_min(Λ') (the sublattice) for d²_min(Λ), making the baseline inconsistent with the subsequent distance-gain factor d²_min(C)/d²_min(Λ), which correctly uses Λ. It would be helpful to rewrite the baseline as γ(Λ) = 2^{−ρ(Λ)} d²_min(Λ) so that the distance-gain factor is well-defined relative to a consistent reference.

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### 3. Fundamental Coding Gain Formula Uses State Count Exponent Instead of Redundancy

**Status**: [Pending]

**Quote**:
> The fundamental coding gain $\gamma$ is given by the formula $2^{-e}d_{\mathrm{min}}^2$ and is also given in decibels.

**Feedback**:
The paper defines γ(C) = 2^{−ρ} · d²_min throughout. For the 8-state one-dimensional code (e=3, ρ=2, d²_min=10), the formula 2^{−e}·d²_min = 10/8 = 1.25, whereas the table gives γ = 2.5 = 2^{−2}·10 = 2^{−ρ}·d²_min. The formula 2^{−e}·d²_min coincides with 2^{−ρ}·d²_min only when e = ρ, which holds for the 4-state one-dimensional code but not in general. It would be helpful to rewrite as 2^{−ρ}d²_min to match the paper's own definition.

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### 4. Primal Code Formula Uses φ^n Instead of φ^μ in Leading Term

**Status**: [Pending]

**Quote**:
> If $\Lambda$ is a decomposable $N$-dimensional complex binary lattice of depth $\mu$, with code formula
> 
> $$
> \Lambda = \phi^ {n} \boldsymbol {G} ^ {N} + \phi^ {\mu - 1} C _ {\mu - 1} + \dots + C _ {0},
> $$

**Feedback**:
The leading term should be φ^μ G^N, since the depth-μ condition means the lattice contains all vectors in φ^μ G^N. The remaining terms correctly use μ. The dual formula immediately below writes φ^μ G^N correctly. This appears to be a typographical inconsistency (n vs. μ) that propagates into the verification sentence. It would be helpful to rewrite the leading term as φ^μ G^N throughout.

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### 5. Dual Code Formula and Verification Sentence Use n and u Instead of μ

**Status**: [Pending]

**Quote**:
> This proposition may be verified by noting that $C_0^\perp / \dots / C_{n-1}^\perp$ is a code partition chain, that every generator $\phi^{\mu - j - 1} \pmb{g}_j^\perp$ of $\Lambda^\perp / \phi^n \pmb{G}^N$ is orthogonal mod $\phi^u$ to every generator $\phi^{j'} \pmb{g}_{j'}$ of $\Lambda / \phi^n \pmb{G}^N$

**Feedback**:
The depth parameter throughout Section II-H is μ, yet the verification sentence uses n (for the chain length C_0^⊥/…/C_{n-1}^⊥ and the coset Λ^⊥/φ^n G^N) and u (for the orthogonality modulus). The dual chain has μ levels (C_0^⊥ through C_{μ-1}^⊥), the coset should be Λ^⊥/φ^μ G^N, and the orthogonality condition should be mod φ^μ. It would be helpful to rewrite n→μ and u→μ throughout this sentence for consistency with the rest of Section II-H.

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### 6. Dual Partition Chain Uses φ^n Instead of φ^μ

**Status**: [Pending]

**Quote**:
> then $\pmb{G}^{N} / \Lambda / \Lambda' / \phi^{n}\pmb{G}^{N}$ and $\pmb{G}^{N} / \Lambda'^{\perp} / \Lambda^{\perp} / \phi^{n}\pmb{G}^{N}$ are dual partition chains

**Feedback**:
The depth of the lattices in this paragraph is stated to be μ, and the analogous single-lattice chain earlier in the section is written G^N/Λ/φ^μ G^N. Consistency requires the bottom of the chain to be φ^μ G^N, not φ^n G^N. This is the same n-vs-μ inconsistency as in the code formula and verification sentence. It would be helpful to rewrite both occurrences of φ^n G^N as φ^μ G^N.

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### 7. Self-Duality Claim for D₄ Requires Explicit Justification Under Complex Inner Product

**Status**: [Pending]

**Quote**:
> For example, the Schlafli lattice $D_{4}$ is self-dual, because its complex code formula is $D_{4} = \phi G^{2} + (2,1,2)$, and the $(2,1,2)$ code is self-dual.

**Feedback**:
Under the standard binary inner product, the (2,1,2) repetition code {00,11} has dual equal to the full (2,2) code, not (2,1,2) itself—since (11,11)=0 mod 2 makes it self-orthogonal but not self-dual. The paper's own real-lattice analysis states D₄ and RD₄ are dual to each other, not self-dual. The self-duality claim for D₄ under the complex definition requires an explicit argument distinguishing the complex inner product mod φ from the standard binary inner product, and reconciling the apparent conflict with the real-lattice duality statement.

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### 8. E₈ Self-Duality Requires a Non-Trivial Lattice Identity That Is Left Implicit

**Status**: [Pending]

**Quote**:
> The Gosset lattice $E_{8}$ is self-dual, because its complex code formula is $E_{8} = \phi^{2}G^{4} + \phi (4,3,2) + (4,1,4)$, and the $(4,3,2)$ and $(4,1,4)$ codes are duals.

**Feedback**:
Applying the dual construction (reverse chain, dualize each code): Λ^⊥ = φ²G⁴ + φ·(4,1,4)^⊥ + (4,3,2)^⊥ = φ²G⁴ + φ·(4,3,2) + (4,1,4). This matches the original E₈ formula only if swapping the (4,3,2) and (4,1,4) codes at different depth levels yields the same lattice—a non-trivial identity. Readers might note that this lattice equality should be derived or cited explicitly, rather than left as an implicit consequence of the code duality alone.

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### 9. Complex Code Formula for H₁₆ in Table I Has Wrong Superscript on G

**Status**: [Pending]

**Quote**:
> 16} + (16,15,2) | φG^{8} + (8,7,2)  |
> |  (1,3) | H_{16} | 16 | 2 | 2Z^{16} + (16,11,4) | φ^{2}G^{4} + φ(8,7,2) + (8,4,4)  |
> |  (0,3) | Λ_{1

**Feedback**:
For H₁₆ = Λ(1,3): n=3, r=1, N=2³=8. The general formula gives φ^{n−r}G^N = φ²G^8. The table entry writes φ²G^4, but N=8 requires G^8. Checking adjacent entries confirms the pattern: E₈=Λ(0,2) uses G^4 (N=4), D₄=Λ(0,1) uses G^2 (N=2). For H₁₆ with N=8, the correct entry should be φ²G^8 + φ(8,7,2) + (8,4,4).

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### 10. Duplicate Row for H₁₆ in Table II

**Status**: [Pending]

**Quote**:
> |  H_{16} | 11 | 11/8 | 5 | 5/8 | 4 | 2^{11/8} | 4.14 | 284 | 8 | ~32  |
> |  H_{16} | 11 | 11/8 | 5 | 5/8 | 4 | 2^{11/8} | 4.14 | 284 | 8 | ~32  |

**Feedback**:
Table II contains two identical rows for H₁₆ with all parameters equal (k=11, κ=11/8, r=5, ρ=5/8, d²_min=4, γ=2^{11/8}, 4.14 dB, N̂₀=284, 2^ν=8, Ñ_D≈32). In the 16-dimensional Barnes–Wall family the relevant intermediate lattice between D₁₆ and Λ₁₆ is H₁₆ alone; the 32-dimensional table correctly lists both X₃₂ and H₃₂ as distinct entries. The second H₁₆ row should either be removed or replaced with the correct distinct intermediate lattice and its parameters.

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### 11. Minimum Hamming Distance Argument Does Not Rule Out Two-Step Re-Merge

**Status**: [Pending]

**Quote**:
> The minimum Hamming distance of this code (taking the outputs as $c(a)$) is five, because the distance between distinct paths is at least two where they diverge, two where they merge, and one somewhere in between.

**Feedback**:
The argument 'at least two where they diverge, two where they merge, and one somewhere in between' implicitly assumes the shortest error event spans at least three time steps. For a two-step re-merge (diverge at t, re-merge at t+1), the path difference would have weight 2+2=4, not 5. The paper does not explicitly rule out two-step re-merges for this specific encoder (g₀=[10], g₁=[11]). It would be helpful to verify that no two-step re-merge is possible by checking the state transitions explicitly, since the claim d_H=5 depends on this.

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### 12. Regularity Condition (b) Cites a Four-Level Chain with Repeated Final Exponent

**Status**: [Pending]

**Quote**:
> b) if it is an Ungerboeck labeling and the Ungerboeck distance bound always holds with equality, e.g., any Ungerboeck labeling for $Z^2 / RZ^2 / 2Z^2 / 2RZ^2 = G / \phi G / \phi^3 G / \phi^3 G$;

**Feedback**:
The chain G/φG/φ²G/φ³G corresponds to Z²/RZ²/2Z²/2RZ², with exponents 0,1,2,3. The paper writes the last two entries as φ³G/φ³G, which is a trivial (order-1) partition. The correct chain should be G/φG/φ²G/φ³G, with the third step being φ²G (not φ³G). The E₈ example confirms: d_H(C_0)=4, d_H(C_1)=2, consistent with exponents 0,1,2,3. It would be helpful to rewrite as G/φG/φ²G/φ³G.

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### 13. Subcode Constraint K − K' ≤ K' Lacks Justification from Subcode Relationship

**Status**: [Pending]

**Quote**:
> The generators $\{2g_{k}, K' + 1 \leq k \leq K\}$ generate a lattice $\Lambda'$ that is a sublattice of $\Lambda_{e}$, whose elements are congruent to $2c'$ modulo 4, where $c'$ is a codeword in a binary $(N, K - K')$ block code $C'$ that is a subcode of the code $C$, so $K - K' \leq K'$.

**Feedback**:
The inequality K−K' ≤ K' (equivalently K ≤ 2K') is stated as a consequence of C' being a subcode of C, but a subcode relationship C'⊆C only gives dim(C') ≤ dim(C), i.e., K−K' ≤ K, not K−K' ≤ K'. To obtain K−K' ≤ K', one would need C' to be a subcode of C₁ (dimension K'), not merely of C. It would be helpful to either establish that the generators g_k for k>K' lie in C₁, or correct the stated bound.

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### 14. Hamming Distance Ratio '2/4/…' Is Ambiguous and Appears in the Wrong Order

**Status**: [Pending]

**Quote**:
> This suggests that we shall want to choose a code partition chain $C_{\mu - 1} / C_{\mu - 2} / \dots / C_0$ for which the Hamming distances are in the ratio $2 / 4 / \dots$.

**Feedback**:
The minimum squared distance formula d²_min = min[2^μ, 2^{μ−1}d_H(C_{μ−1}), …, d_H(C_0)] is equalized when d_H(C_j) = 2^{μ−j}, so d_H(C_0):d_H(C_1):…:d_H(C_{μ−1}) = 2^μ:2^{μ−1}:…:2. The distances decrease as j increases. The stated ratio '2/4/…' is ambiguous and appears to be in the wrong order relative to the chain C_{μ−1}/…/C_0. The E₈ example confirms: d_H(C_0)=4, d_H(C_1)=2, ratio 4:2 for C_0:C_1. It would be helpful to rewrite as 'd_H(C_0):d_H(C_1):…:d_H(C_{μ−1}) = 2^μ:2^{μ−1}:…:2'.

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### 15. Order of φ^μ G Sublattice Written with Typographically Ambiguous Exponent

**Status**: [Pending]

**Quote**:
> More generally, $\phi^{\mu}G$ is a sublattice of $G$ of order $|\phi|^2\mu = 2^\mu$

**Feedback**:
The expression |φ|²μ is typographically ambiguous between the product 2·μ and the power 2^μ. The correct value is |φ^μ|² = |φ|^{2μ} = 2^μ. It would be helpful to rewrite as |φ^μ|² = 2^μ to unambiguously express the norm of φ^μ.

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### 16. Gaussian Primes of Norm 5 Are Incomplete: 1±2i Omitted

**Status**: [Pending]

**Quote**:
> The primes of $G$, in order of increasing norm, are $1 + i$, $2 \pm i, 3, \dots$, with norms $2, 5, 9, \dots$.

**Feedback**:
The Gaussian primes of norm 5 include both 2±i and 1±2i (since |1+2i|²=5 and (2+i)/(1+2i) is not a unit). The list '2±i' covers only two of the four norm-5 Gaussian primes; 1±2i are omitted. Readers might note that a complete list would read '2±i, 1±2i, 3, …, with norms 2, 5, 5, 9, …' or alternatively note that there are four Gaussian primes of norm 5.

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### 17. Lemma 2 Statement Uses Approximation Symbol Where Equality Is Intended

**Status**: [Pending]

**Quote**:
> Then there is a sequence of lattices $\Lambda_0 = \Lambda$, $\Lambda_1, \dots, \Lambda_K \approx \Lambda'$ such that $\Lambda_0 / \Lambda_1 / \dots / \Lambda_K$ is a lattice partition chain

**Feedback**:
The proof sketch explicitly states 'Λ_K = Λ' is certainly such a lattice,' using equality. The approximation symbol '≈' in the lemma statement is almost certainly a typographical artifact for '='. The partition chain Λ_0/Λ_1/…/Λ_K requires Λ_K to be exactly Λ' for the chain to connect Λ to Λ'. It would be helpful to rewrite Λ_K ≈ Λ' as Λ_K = Λ'.

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### 18. Inductive Step Does Not Verify That Λ_k Is a Binary Lattice

**Status**: [Pending]

**Quote**:
> Let $\Lambda_k$ then be the union of $\Lambda_{k + 1}$ and its coset $\Lambda_{k + 1} + \mathbf{g}_k$; $\Lambda_k$ is clearly a lattice, which by construction is a sublattice of $\Lambda$ and has $\Lambda_{k + 1}$ (and thus $\Lambda'$ ) as a sublattice.

**Feedback**:
The condition g_k + g_k ∈ Λ_{k+1} ensures 2g_k ∈ Λ_{k+1}, but the proof does not verify that 2λ ∈ Λ_{k+1} for all λ ∈ Λ_k (only for the generator g_k). Without this, the induction does not establish that each intermediate Λ_k is a binary lattice, which is needed for the chain Λ_0/Λ_1/…/Λ_K to consist of binary lattices as claimed. It would be helpful to add a sentence confirming that 2(Λ_{k+1} + g_k) = 2Λ_{k+1} + 2g_k ⊆ Λ_{k+1} follows from 2g_k ∈ Λ_{k+1} and Λ_{k+1} being binary.

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### 19. Wei-Type Code Definition Has Density Direction Reversed

**Status**: [Pending]

**Quote**:
> We shall say that a code is "Wei-type" if $\Lambda / \Lambda'$ is a lattice partition where $\Lambda'$ is a denser lattice than $\Lambda$.

**Feedback**:
As a sublattice, Λ' ⊆ Λ necessarily has larger fundamental volume and lower point density than Λ. For example, in Z^8/E_8, E_8 as a sublattice of Z^8 has V(E_8) = 16·V(Z^8), so E_8 is sparser than Z^8 in this embedding. What Wei stresses is that Λ' should be a densely-packed lattice (high minimum distance relative to its own fundamental volume), not that it is denser than Λ in absolute terms. It would be helpful to rewrite as 'where Λ' is a sublattice of Λ chosen to be a densely-packed lattice.'

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### 20. State Count Column Header '2°' Is Undefined

**Status**: [Pending]

**Quote**:
> |  N | Λ | Λ' | 2° | k/(k+r) | ρ | d^{2}_{min} | γ | dB | N_{0} | N_{D} | γ_{eff}(dB)  |

**Feedback**:
The column header '2°' appears in Tables VII–XI to represent the number of encoder states (2^ν), but the superscript '°' is not defined anywhere in the paper. The paper uses 2^ν for encoder states throughout Sections V–VI. This ambiguity is consequential because the folk theorem in Section VII is stated in terms of state counts, and readers cannot verify the cross-class comparisons without knowing which exponent '°' represents. It would be helpful to rewrite the column header as '2^{ν}' with ν defined in the caption.

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### 21. Trellis Diagram Description Conflates State Count 2^r with 2^ν

**Status**: [Pending]

**Quote**:
> A trellis diagram for a $2^{r}$-state, rate-$k / (k + r)$ convolutional code is an extended state transition diagram for the encoder that generates $C$. For each time $t$, it has $2^{\nu}$ nodes, or states, representing the possible states at time $t$.

**Feedback**:
The section heading calls the code '2^r-state' but the trellis is described as having 2^ν nodes, where ν is the overall constraint length. These are generally distinct: a rate-1/2 code (r=1) can have ν=2 (4 states) or ν=3 (8 states). The paper's own definition states 'such an encoder has 2^ν states.' The phrase '2^r-state, rate-k/(k+r) convolutional code' conflates r with ν. It would be helpful to rewrite as '2^ν-state, rate-k/(k+r) convolutional code.'

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### 22. Shape Gain γ_s Is Defined Circularly

**Status**: [Pending]

**Quote**:
> The total coding gain $\gamma_{\text{tot}}(\mathcal{C})$ is the product of the fundamental coding gain $\gamma(\mathcal{C})$ with the shape gain $\gamma_s$ of the finite constellation ($\gamma_s$ is defined as $\gamma_{\text{tot}}(\mathcal{C})/\gamma(\mathcal{C})$ and is approximately equal to the ratio of the normalized second moment [7] of an $N$-cube to that of the region of $N$-space in which the constellation is contained).

**Feedback**:
Defining γ_s = γ_tot/γ(C) is circular: γ_tot is introduced as γ(C)·γ_s, and then γ_s is immediately defined as γ_tot/γ(C). The substantive content is the approximation via normalized second moments. The logical order should be reversed: first define γ_s independently as the ratio of normalized second moments, then define γ_tot = γ(C)·γ_s as a consequence.

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### 23. Constellation Expansion Factor Omits Lattice Redundancy Contribution

**Status**: [Pending]

**Quote**:
> m a different coset of  $\Lambda'$ . Given a selected coset of  $\Lambda'$ ,  $n - k$  uncoded bits select a particular signal point from that coset. The constellation expansion factor (compared to an uncoded constellation of  $2^n$  points from a coset of  $\Lambda$ ) is thus  $2^{r(C)}$  per  $N$  dimensions, or  $2^{\rho(C)}$  per two dimensions. This translates into an average power cost of a factor of  $2^{\rho(C)}$  (or  $\rho(C) \cdot 3.01 \, \mathrm{dB}$ ), which is reflected in the formula for the fundamental coding gain  $\gamma(\mathbb{C})$  just given.

**Feedback**:
The constellation expansion factor is stated as 2^{r(C)} (encoder redundancy only), but the fundamental coding gain formula uses ρ(C) = 2r(C)/N where r(C) includes both encoder and lattice redundancy. When Λ has nonzero redundancy (e.g., D₄ or E₈), the actual power cost relative to an uncoded Z^N-based system includes both contributions. The verbal explanation mentions only the encoder redundancy, creating an inconsistency with the formula. It would be helpful to rewrite to state the expansion factor as 2^{r(C)+r(Λ)} relative to an uncoded Z^N constellation, or to clarify that r(C) here already subsumes the lattice redundancy.

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### 24. Linearity Condition (a) Overstates When Regularity Holds for Mod-2 Lattices

**Status**: [Pending]

**Quote**:
> a) if it is linear, in the sense that $c(\pmb{a}) - c(\pmb{a}') = c(\pmb{a} \oplus \pmb{a}')$ modulo $\Lambda'$, e.g., whenever $\Lambda$ and $\Lambda'$ are mod-2 lattices;

**Feedback**:
The claim that the labeling is automatically regular 'whenever Λ and Λ' are mod-2 lattices' overstates the result. Linearity of the labeling map c(a) is an additional assumption on the labeling, not a consequence of Λ and Λ' being mod-2 lattices—a mod-2 partition can in principle be labeled non-linearly. The paper itself notes earlier that 'a trellis code has little chance of being linear unless the mapping c(a) … is linear modulo Λ'.' It would be helpful to rewrite as 'e.g., whenever Λ and Λ' are mod-2 lattices and c(a) is chosen to be a linear labeling modulo Λ'.'

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### 25. Folk Theorem State-Count Thresholds Contradicted by Table XI Without Quantitative Qualification

**Status**: [Pending]

**Quote**:
> In summary, we propose a folk theorem: it takes two states to get 1.5 dB, four states to get 3 dB, 16 states to get 4.5 dB, perhaps 64 states to get 5.25 dB, and 256 states to get 6 dB, as long as we require a reasonably small error coefficient (for trellis codes).

**Feedback**:
The paper's own Table XI shows the Class V H₁₆/Λ₁₆ code achieves γ = 2^{9/4} ≈ 6.77 dB with only 64 states, and the Class I Λ₁₆/RΛ₁₆ code achieves 6 dB with 16 states—both apparently violating the folk theorem's implied monotonicity. These cases are dismissed due to large error coefficients, but no quantitative threshold for 'reasonably small' Ñ₀ is provided, leaving the theorem's domain of validity undefined. It would be helpful to restate the folk theorem as a conditional result with an explicit bound on Ñ₀ and to note explicitly that it is an empirical observation from the surveyed codes rather than a proven bound.

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