# Coset Codes—Part I: Introduction and Geometrical Classification

**Date**: 03/08/2026
**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.

**Eight-Class Taxonomy Is Presented as Complete but Not Proven Exhaustive**

Section VI introduces eight generic code classes defined by three binary design choices, and Section VII draws broad conclusions—including a 'folk theorem' about state complexity versus coding gain—as though this taxonomy covers the full design space of binary lattice coset codes. However, the paper never formally argues that these eight classes are exhaustive. Codes with non-uniform partition orders, mixed-depth partition chains, rate-k/(k+r) encoders with r > 2, or time-varying convolutional encoders could fall outside all eight classes without violating any stated definition, and they are neither subsumed nor ruled out as potentially superior. The folk theorem is stated as a conclusion from the enumerated classes rather than as a consequence of any capacity or sphere-packing bound applicable to all binary coset codes. It would be helpful to either provide a formal argument—perhaps via Lemma 5's equivalence to Z^N/2^m Z^N codes—bounding the parameter space and showing no qualitatively distinct class is omitted, or to reframe the folk theorem explicitly as a conjecture supported by the enumerated examples rather than a conclusion derived from a complete classification. Without this, readers may incorrectly interpret the taxonomy as a completeness theorem, which could misdirect subsequent search efforts.

**Effective Coding Gain Rule of Thumb Lacks Rigorous Justification and May Distort Code Comparisons**

Throughout Sections V–VII and Tables IV–XI, the paper converts error coefficients into effective coding gain penalties using the rule that 'every factor of two increase in the error coefficient reduces the coding gain by about 0.2 dB.' This rule is introduced without derivation, without citation, and without specifying the SNR regime or target error probability (e.g., 10^{-6}) at which it is calibrated. The union bound on word-error probability depends on the full distance spectrum, not just the nearest-neighbor multiplicity Ñ_0, and the 0.2 dB/factor-of-two approximation can be substantially off when higher-distance terms dominate—a situation the paper itself acknowledges by marking dominant coefficients with asterisks in Table V. The correction is also applied inconsistently: it is omitted entirely for lattice codes in Section VII and for several Class V and VI entries where Ñ_0 is unknown or very large (e.g., the 8-state E_8/RE_8 code with Ñ_0 = 764, or the Class VI Λ_16/RΛ_16 code). Because γ_eff is the primary metric used in Figure 12 to rank codes across classes and to support the conclusion that 'Ungerboeck codes are still the benchmark,' it would be helpful to state the precise error-probability model underlying the rule, to bound the approximation error from truncating the distance spectrum at the first term, and to flag explicitly which comparisons in Figure 12 could be affected by missing or extreme error coefficients.

**Minimum Squared Distance Formula for Decomposable Mod-4 Lattices Requires a Gap-Filling Argument**

Section II-G gives the minimum squared distance of a decomposable mod-4 real lattice as d²_min(Λ) = min[16, 4d_H(C_1), d_H(C_0)], and the analogous formula for mod-2 lattices appears in Lemma 3. Both derivations implicitly assume that the minimum norm within each nonzero coset of 4Z^N is achieved by the coset representative of minimum Hamming weight—an identification that holds when generators g_k are binary (0/1) vectors but can fail when they take values in {0, ±1, 2} or are integers modulo 4, because a coordinate with a nonzero ones-coefficient can simultaneously carry a nonzero twos-coefficient from a carry, potentially altering the norm. The paper acknowledges that indecomposable lattices (e.g., Λ_24) require separate treatment, but does not verify that the decomposability condition is both necessary and sufficient to prevent this issue in all cases used in the paper. Since Table I applies these formulas to Λ_24 and H_24 with prime notation indicating non-standard generators, and since the coding gain calculations in Tables VI–XI depend directly on these distance values, it would be helpful to state precisely which lattices satisfy the conditions under which the distance formulas are valid, and to add a brief verification—by explicit case analysis on the carry structure or by a precise forward reference to Part II—that the formula holds for every lattice appearing in the trellis code constructions.

**Dual Lattice Construction Introduces an Unresolved Asymmetry Between Real and Complex Duality Conventions**

Section II-H defines the dual lattice Λ^⊥ using the complex (Hermitian) inner product modulo φ^μ, and states that for real lattices with odd φ-depth μ = 2m−1, the real dual differs from the complex dual by a rotation R. This equivalence is asserted without proof. The rotation R includes a reflection (det R = −2 in two dimensions), so RΛ and Λ are not generally isometric, and it is not obvious that the duality relationship is preserved under this rotation. The paper states that 'for the lattices that we will be considering' the two definitions coincide when μ is even, but does not verify this claim for each lattice in Table I. Notably, D_4 has μ = 1 (odd) and is described as self-dual under the complex definition yet has real dual RD_4—a discrepancy that is not fully resolved. Partition chains in Table III mix lattices of odd and even depth (e.g., D_16/H_16/Λ_16 has depths 1/2/3), so the dual partition chain involves lattices of both parities, and the duality relationships used in Section III and in the Class V and VI constructions of Section VI depend on which convention is in force at each level. It would be helpful to supply a brief proof that the two duality definitions agree up to the stated rotation for all lattices used in the paper, or to add a reconciliation table clarifying which notion applies at each level of each partition chain, so that the partition depth and order calculations in Table III are verifiably internally consistent.

**Trellis Decoding Complexity Metric Is Implementation-Specific and Inconsistently Comparable Across Code Classes**

Tables II, III, and IV–XI report normalized decoding complexity Ñ_D as the total number of binary operations for the trellis-based partition decoder (from Part II) plus the Viterbi algorithm cost. The paper acknowledges this measure is 'highly implementation-dependent,' yet uses it as the primary x-axis variable in the performance-versus-complexity plots of Figure 12, from which the main comparative conclusions of Section VII are drawn. Two specific consistency issues arise: first, the partition decoding cost and the Viterbi cost are added without justification that they scale similarly with dimension N or encoder state count; second, for partitions of different dimensions (e.g., Z²/2RZ² at N=2 versus Z^8/E_8 at N=8), normalization to two dimensions conceals fundamentally different trellis structures. The entry for the Class VI Λ_16/RΛ_16 code with Ñ_D ≈ 2^19 dwarfs all other entries and suggests the metric has broken down as a useful comparator at large partition orders. Additionally, several Table III entries for large-order partitions yield the ratio Ñ_D/|Λ/Λ'| slightly above the stated upper bound of 6 (e.g., H_16/Λ_16 gives α ≈ 6.06), and the source of the trellis-based counts is deferred to Part II, making independent verification difficult. It would be helpful to either separate the two complexity contributions in the tables, state explicitly the assumptions under which the additive model is valid, and note the range of uncertainty in the cross-class comparisons, so that the conclusions about trellis codes versus lattice codes in Section VII rest on a uniform and verifiable basis.

**Coding Gain Formula and Capacity Gap: Missing Information-Theoretic Anchor**

The fundamental coding gain γ(C) = 2^{−ρ(C)} d²_min(C) is the central performance metric throughout the paper, but it captures only minimum squared Euclidean distance and constellation expansion penalty, omitting the error coefficient Ñ_0 that can vary by three orders of magnitude across codes with identical γ. More broadly, the paper provides no information-theoretic bound—such as a sphere-packing bound, a Varshamov-Gilbert-type bound, or a capacity-approaching condition—that would allow a reader to assess how close any classified code comes to the Shannon limit for the AWGN channel at the corresponding spectral efficiency. The 'folk theorem' in Section VII is explicitly heuristic, and without at least one reference point connecting γ(C) or γ_eff to the gap to capacity, it is impossible to judge whether the approximately 6 dB ceiling observed across all 256-state codes reflects a fundamental barrier, a limitation of the coset code framework, or an artifact of the families searched. Furthermore, the coding gain formula for trellis codes rests on an informal density argument (Section IV-B) that assumes the time-zero lattice Λ_0 is the same for all starting states—verified for linear codes but not for the nonlinear codes also covered in Section IV-C. It would be helpful to include a brief comparison of the best γ_eff values in Figure 12 against the capacity limit at the corresponding normalized redundancy ρ, and to either restrict the coding gain formula explicitly to linear trellis codes or provide the additional argument needed to extend it to the regular nonlinear codes discussed in Section IV-C.

**Uniform MSED across cosets assumed but not guaranteed for non-standard partition chains**

Assumption involved: The paper's distance analysis for trellis codes (Section IV-B and the Class I–VIII code constructions) relies on the premise that the minimum squared Euclidean distance (MSED) within every nonzero coset of Λ' in the partition Λ/Λ' is the same value d²_min(Λ'), and that the number of coset points achieving this minimum (N_Λ in the error-coefficient formula N₀ = N_{Λ'} + (2^k − 1)N²_Λ) is identical across all nonzero cosets.

Where the tension arises: This uniformity holds cleanly for the Barnes–Wall family and their principal sublattices because those lattices are highly symmetric (self-dual, transitive automorphism groups). However, the paper also introduces partitions involving non-standard or 'irregular' sublattices—most notably the indecomposable Leech lattice Λ₂₄ and its half-lattice H₂₄ (Table I, marked with primes), for which the paper itself notes that the standard code-formula notation breaks down and must be replaced by a modified convention. For such lattices the coset weight distributions are not guaranteed to be coset-uniform, so the single value N_Λ used in the error-coefficient formula may not apply equally to all nonzero cosets.

Consequence: If coset MSEDs or multiplicities vary across cosets, the error coefficient N₀ computed from the uniform formula will be an underestimate for some transmitted sequences and an overestimate for others, making the effective coding gain γ_eff unreliable for those codes. The paper does not provide a coset-by-coset verification for the Leech-lattice-based entries in Tables I–III.

Suggested check: It would be helpful to verify, for each partition Λ/Λ' involving Λ₂₄ or H₂₄, that the minimum norm and its multiplicity are indeed constant across all nonzero cosets of Λ', or to flag those entries in the tables where this uniformity has not been confirmed.

**Exhaustiveness claim for known code classes rests on Lemma 5's equivalence, which requires the labeling to be binary and linear**

Assumption involved: Lemma 5 (Section IV-B) is used to argue that 'all trellis codes based on partitions of binary lattices are equivalent to trellis codes based on partitions G^N/φ^μG^N … so that in principle only these kinds of partitions need to be considered to discover all binary trellis codes.' The paper then uses this as a basis for the claim in Section VII that a 'systematic search of Z^N/4Z^N codes should settle the question' of whether significantly better codes exist.

Where the tension arises: The proof of Lemma 5 constructs the augmented encoder C' by appending k(Λ') uncoded bits that select cosets of φ^μG^N within Λ'. This construction is valid only when the labeling of the partition Λ'/φ^μG^N is itself binary and admits a consistent Ungerboeck-style decomposition into two-way partitions (Lemma 2). For indecomposable mod-4 lattices such as Λ₂₄ (explicitly noted as indecomposable in Section II-G), the generators of Λ'/φ^μG^N involve integers modulo 4 that cannot be expressed purely in terms of binary codes, so the 'uncoded bits' a' in the augmented encoder are not straightforwardly binary. The paper acknowledges this for Λ₂₄ but does not revisit the scope of Lemma 5's exhaustiveness claim.

Consequence: Codes built on indecomposable partitions may not be fully captured by the Z^N/4Z^N search space implied by Lemma 5, leaving open the possibility that coset codes with non-standard partition chains (e.g., those involving Λ₂₄ directly as Λ') could achieve parameters outside the classification presented in Sections V–VI.

Suggested check: Readers might note that it would be helpful to either extend Lemma 5 explicitly to cover indecomposable lattices (with a careful treatment of the non-binary generators) or to qualify the exhaustiveness conclusion in Section VII to apply only to the decomposable case.

**Status**: [Pending]

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## Detailed Comments (25)

### 1. R Described as Rotation but Has Determinant −2, Making It an Improper Rotation

**Status**: [Pending]

**Quote**:
> The most important scaled orthogonal transformation for our purposes is the rotation operator $R$, defined by the $2 \times 2$ matrix
> 
> $$
> R \triangleq \left( \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right).
> $$
> 
> $R\mathbb{Z}^2$ is a version of $\mathbb{Z}^2$ obtained by rotating $\mathbb{Z}^2$ by $45^\circ$ and scaling by $2^{1/2}$

**Feedback**:
Computing det(R) = (1)(−1) − (1)(1) = −2, so R is a scaled improper rotation (rotation composed with a reflection), not a pure rotation. A scaled rotation by angle θ and scale s has det = s² > 0; here det = −2 < 0. This distinction matters in the duality arguments of Section II-H, where the sign of the determinant affects whether RΛ and Λ share the same dual lattice. It would be helpful to rewrite 'rotation operator' as 'scaled improper rotation (rotation-reflection) operator' and replace 'rotating Z² by 45° and scaling by √2' with 'applying a 45° rotation composed with a reflection and scaling by √2.'

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### 2. Cartesian Product Density Comparison References Same Lattice Twice

**Status**: [Pending]

**Quote**:
> ^M) = d_{\min}^2(\Lambda)$ and $V(\Lambda^M) = V(\Lambda)^M$. Thus if $\Lambda_N$ is an $N$-dimensional lattice and $\Lambda_M$ is an $M$-dimensional lattice, then in $MN$-space (where they can be compared<!-- PAGE BREAK -->FORNEY: COSET CODES—PART I
> 1129
> 
> directly) $\Lambda_N^M$ has a greater or lesser density of points per unit volume than does $\Lambda_N^M$ according to whether $\gamma(\Lambda_N)$ is greater or less than $\gamma(\Lambda_M)$, provided that they are scaled so that their minimum squared distances are the same. We therefore say that $\Lambda$ is denser than $\Lambda'$ if $\gamma(\Lambda) > \gamma(\Lambda')$

**Feedback**:
The second occurrence of Λ_N^M should be Λ_M^N. The M-fold product of the N-dimensional lattice lives in MN dimensions with V(Λ_N^M) = V(Λ_N)^M, and the N-fold product of the M-dimensional lattice lives in the same MN dimensions with V(Λ_M^N) = V(Λ_M)^N. Comparing these two at equal minimum squared distance gives Λ_N^M denser iff γ(Λ_N) > γ(Λ_M), which is the intended claim. As written, the sentence compares a lattice to itself, which is trivially equal. Rewrite the second '$\Lambda_N^M$' as '$\Lambda_M^N$.'

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### 3. Scaling Invariance of Coding Gain Uses Incorrect Distance Formula for General Scaled Orthogonal T

**Status**: [Pending]

**Quote**:
> the fundamental coding gain is invariant to any scaled orthogonal transformation $T$, $\gamma(T\Lambda) = \gamma(\Lambda)$, because $d_{\min}^2(T\Lambda) = |\det T|^{2/N} d_{\min}^2(\Lambda)$ and $V(T\Lambda) = |\det T| V(\Lambda)$, where $\det T$ is the determinant of $T$.

**Feedback**:
The formula d²_min(TΛ) = |det T|^{2/N} d²_min(Λ) holds only when T = cQ with Q orthogonal and c a scalar, because then ||Tλ||² = c²||λ||² and |det T| = |c|^N, giving |det T|^{2/N} = c². For a general non-isotropic scaled orthogonal transformation the minimum distance can be much smaller than |det T|^{2/N} d²_min(Λ). Since R is a specific scalar multiple of an orthogonal matrix, the invariance claim is correct for all cases actually used in the paper, but the stated formula is not valid for arbitrary scaled orthogonal T. It would be helpful to rewrite '$d_{\min}^2(T\Lambda) = |\det T|^{2/N} d_{\min}^2(\Lambda)$' as '$d_{\min}^2(T\Lambda) = c^2 d_{\min}^2(\Lambda)$, where $T = cQ$ with $Q$ orthogonal and $c > 0$.'

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### 4. Coset Decomposition Formula for Partition Chain Has Reversed Summation Order Relative to Concrete Example

**Status**: [Pending]

**Quote**:
> $$
> \Lambda = \Lambda'' + \left[ \Lambda' / \Lambda'' \right] + \left[ \Lambda / \Lambda' \right],
> $$
> 
> meaning that every element of $\Lambda$ can be expressed as an element of $\Lambda''$ plus a coset representative from $[\Lambda' / \Lambda'']$ plus a coset representative from $[\Lambda / \Lambda']$. For example, the chain $Z / 2Z / 4Z / \cdots$ leads to the standard binary representation of an integer $m$:
> 
> $$
> m = a_0 + 2a_1 + 4a_2 + \cdots
> $$
> 
> where $a_0, a_1, a_2, \dots \in \{0,1\}$, and $a_0$ specifies the coset in the partition $Z / 2Z$, $2a_1$ specifies the coset in the partition $2Z / 4Z$, and so forth. That is,
> 
> $$
> Z = \left[ Z / 2Z \right] + \left[ 2Z / 4Z \right] + \left[ 4Z / 8Z \right] + \cdots.
> $$

**Feedback**:
The general formula places [Λ/Λ'] last (coarsest partition last), but the concrete binary expansion Z = [Z/2Z] + [2Z/4Z] + ⋯ writes [Z/2Z] first (coarsest partition first). In the chain Λ/Λ'/Λ'' with Λ=Z, Λ'=2Z, Λ''=4Z, the general formula gives Z = 4Z + [2Z/4Z] + [Z/2Z], so [Z/2Z] is last—yet the displayed binary expansion writes it first. This reversal is internally consistent but is not flagged for the reader, creating a genuine risk of confusion when applying the formula to the partition tower of Fig. 5(a). It would be helpful to add a note that the summation order in the displayed binary expansion reverses the order of the general formula.

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### 5. Coset Digit φa₁ Specifies Wrong Partition Level

**Status**: [Pending]

**Quote**:
> where $a_0, a_1, a_2, \dots \in \{0,1\}$, and $a_0$ specifies the coset of $\phi G$ in the partition $G / \phi G$, $\phi a_1$ specifies the coset of $\phi^2 G$ in the partition $\phi G / \phi^3 G$, and so forth.

**Feedback**:
The term φa₁ should specify the coset of φ²G in the partition φG/φ²G, not φG/φ³G. In the chain G ⊃ φG ⊃ φ²G ⊃ φ³G ⊃ ⋯, each two-way partition is φ^k G/φ^{k+1}G. The digit φa₁ (with a₁ ∈ {0,1}) selects between 0 and φ within φG, i.e., it specifies the coset of φ²G in φG/φ²G. The analogous real statement has 2a₁ specifying the coset of 4Z in 2Z/4Z, not in Z/4Z. Rewrite 'the partition $\phi G / \phi^3 G$' as 'the partition $\phi G / \phi^2 G$.'

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### 6. Order of φ^μ G Stated with Malformed Intermediate Expression

**Status**: [Pending]

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

**Feedback**:
The correct intermediate expression is |φ^μ|² = (|φ|²)^μ = 2^μ, applying Lemma 1 with g = φ^μ. The text writes |φ|²μ, which reads as the product 2μ (a linear function of μ) rather than 2^μ (an exponential). While the final answer 2^μ is correct, the intermediate notation is ambiguous or wrong. Rewrite '|φ|²μ = 2^μ' as '|φ^μ|² = (|φ|²)^μ = 2^μ.'

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### 7. R² = 2 Claim Contradicts the Matrix Definition of R

**Status**: [Pending]

**Quote**:
> The correspondence is not exact because $R$ includes a reflection as well as rotation and scaling, so that $R^2 = 2$, whereas $\phi^2 = 2i$. (We could have avoided this difficulty by exchanging columns in the definition of $R$.)

**Feedback**:
Computing R² directly from the 2×2 definition: R² = [[1,1],[1,−1]]·[[1,1],[1,−1]] = [[2,0],[0,2]] = 2I, which corresponds to multiplication by 2 (a real scalar), not 2i. However, φ² = (1+i)² = 2i corresponds to a 90° rotation and scaling by 2. So R² = 2I (real scaling) while φ² = 2i (complex rotation-scaling)—the stated discrepancy 'R² = 2, whereas φ² = 2i' is actually correct in the sense that R² acts as real multiplication by 2 while φ² acts as complex multiplication by 2i. The paper's phrasing is terse but not wrong; however, it would be helpful to clarify that 'R² = 2' means R² = 2I (the 2×2 identity scaled by 2), to avoid confusion with the scalar 2.

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### 8. Exponent 'n' vs 'μ' Inconsistency in Dual Lattice Code Formula Verification

**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 of the lattice is μ throughout this section, so the top of the partition chain should be φ^μ G^N, the code partition chain should run C_0^⊥/…/C_{μ−1}^⊥, and the quotients should be Λ^⊥/φ^μ G^N and Λ/φ^μ G^N. The text uses n in three places where μ is required, creating an internal inconsistency with the surrounding notation. (The 'mod φ^u' may be an OCR artifact for 'mod φ^μ' and is noted separately.) Rewrite 'C_0^⊥/…/C_{n−1}^⊥' as 'C_0^⊥/…/C_{μ−1}^⊥', 'Λ^⊥/φ^n G^N' as 'Λ^⊥/φ^μ G^N', and 'Λ/φ^n G^N' as 'Λ/φ^μ G^N.'

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### 9. E₈ Self-Duality Example Incorrectly States That (4,3,2) and (4,1,4) Are Each Self-Dual

**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**:
The (4,3,2) and (4,1,4) codes are duals of each other, not individually self-dual. The dual lattice formula reverses the code chain, giving Λ^⊥ = φ²G^4 + φ(4,1,4) + (4,3,2), which differs from E₈ = φ²G^4 + φ(4,3,2) + (4,1,4) by swapping the two codes. For E₈ to be self-dual one must separately verify that the lattice constructed from the swapped chain is isomorphic to E₈. The phrasing 'the (4,3,2) and (4,1,4) codes are duals' implies each is self-dual, which is false. Rewrite as 'the (4,3,2) and (4,1,4) codes are duals of each other, so the dual lattice formula gives Λ^⊥ = φ²G^4 + φ(4,1,4) + (4,3,2), which is isomorphic to E₈.'

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### 10. Dual Partition Chain Uses φ^n Instead of φ^μ in Two Places

**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, and $\Lambda'^{\perp} / \Lambda^{\perp}$ is a partition of binary lattices with the same order and depth as $\Lambda / \Lambda'$.

**Feedback**:
The depth of the partition is μ throughout this section, and the bottom of both chains should be φ^μ G^N, consistent with the dual lattice definition and the immediately preceding sentence. The use of n here is undefined and inconsistent with the surrounding notation. Rewrite both occurrences of 'φ^n G^N' as 'φ^μ G^N.'

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### 11. H₁₆ Complex Code Formula in Table I Has Wrong Complex Dimension

**Status**: [Pending]

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

**Feedback**:
For H₁₆ = Λ(1,3), r=1, n=3, N=2^{n−1}=8 (complex dimension), depth μ=n−r=2. The general formula gives the leading term φ^{n−r}G^N = φ²G^8, not φ²G^4. The table shows G^4 (N=4 instead of N=8), which is inconsistent with the real dimension 16 = 2N listed in the same row. Additionally, the real code formula '2Z^{16} + (16,11,4)' has depth-1 (mod-2) structure, but μ=2 requires a mod-4 structure: the correct real formula should be '4Z^{16} + 2(16,15,2) + (16,11,4)' analogous to Λ₁₆ = Λ(0,3). Rewrite the H₁₆ row with real code formula '4Z^{16} + 2(16,15,2) + (16,11,4)' and complex code formula 'φ²G^8 + φ(8,7,2) + (8,4,4).'

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### 12. H₃₂ and Λ₃₂ Complex Code Formulas Have Inflated φ Exponents in Table I

**Status**: [Pending]

**Quote**:
> |  (1,4) | H_{32} | 32 | 3 | 4Z^{32} + 2(32,31,2) + (32,16,8) | φ^{5}G^{16} + φ^{2}(16,15,2) + φ(16,11,4) + (16,5,8)  |

**Feedback**:
For H₃₂ = Λ(1,4), r=1, n=4, N=16, depth μ=n−r=3. The general formula gives the leading term φ^{n−r}G^N = φ³G^{16}, not φ⁵G^{16}. The subsequent terms φ²(16,15,2), φ(16,11,4), (16,5,8) are correctly listed after the wrong leading exponent. The same pattern of inflated leading exponents appears in the H₂₄ and Λ₃₂ rows. Rewrite the H₃₂ complex code formula as 'φ³G^{16} + φ²(16,15,2) + φ(16,11,4) + (16,5,8)' and verify the Λ₃₂ and H₂₄ leading exponents against the formula φ^{n−r}G^N.

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### 13. Dual Lattice Formula Gives Λ(0,n)^⊥ = Λ(0,n) but Notation 'Λ(n)' Is Undefined

**Status**: [Pending]

**Quote**:
> Thus $\Lambda(n,n)^{\perp} = G^{N} \simeq Z^{2N}$, $\Lambda(0,n)^{\perp} = \Lambda(n)$, and $\Lambda(n-1,n)^{\perp}, n \geq 1$, is the dual $D_N^{\perp}$ of the checkerboard lattice $D_N$

**Feedback**:
Applying the dual formula with r=0 gives Λ(0,n)^⊥ = φ^n G^N + φ^{n−1}RM(n−1,n) + ⋯ + RM(0,n), which is exactly the code formula for Λ(0,n) itself, confirming self-duality. The notation 'Λ(n)' is undefined in the paper's two-argument indexing scheme and conflicts with the established notation Λ(r,n). Rewrite 'Λ(0,n)^⊥ = Λ(n)' as 'Λ(0,n)^⊥ = Λ(0,n)' (i.e., Λ(0,n) is self-dual).

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### 14. Duplicate Row in Table III for D₈/RE₈ Partition with Inconsistent Orders

**Status**: [Pending]

**Quote**:
> |  D8 | RE8 | 8 | 32 | 3 | 1 | 3/4 | 2 | 8 | 88 | 2.75  |
> |  D8 | RE8 | 8 | 128 | 3 | 1 | 1/4 | 2 | 8 | 280 | 2.2  |

**Feedback**:
Table III contains two rows both labeled D₈/RE₈ with 2N=8 and depth μ=3, but with different orders (32 vs. 128) and different κ values (3/4 vs. 1/4). Since RE₈ is a specific lattice, the partition D₈/RE₈ has a unique order. From Table II, r(D₈)=1 and r(E₈)=4 in 8 dimensions, giving |D₈/E₈| = 2^{4−1} = 8; neither 32 nor 128 matches this, suggesting one row has a mislabeled sublattice (e.g., 2RE₈ or 2D₈). It would be helpful to rewrite the second D₈/RE₈ row with the correct sublattice label to distinguish it from the first.

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### 15. Incorrect Intermediate Expression for V(C) in Displayed Formula

**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 third expression 2^{k+r}V(Λ') contradicts the second expression 2^{−k}V(Λ'): these are equal only if V(Λ') = 0. The intended step is to substitute V(Λ') = 2^{k+r}V(Λ) (from the partition order |Λ/Λ'| = 2^{k+r} and Lemma 1), giving 2^{−k}V(Λ') = 2^{−k}·2^{k+r}V(Λ) = 2^r V(Λ). The chain should read V(C) = V(Λ₀) = 2^{−k(C)}V(Λ') = 2^{r(C)}V(Λ), with the intermediate step V(Λ') = 2^{k+r}V(Λ) stated separately. Rewrite '2^{-k(C)}V(\Lambda') = 2^{k+r}V(\Lambda') = 2^{r(C)}V(\Lambda)' as '2^{-k(C)}V(\Lambda') = 2^{r(C)}V(\Lambda)' (using $V(\Lambda') = 2^{k+r}V(\Lambda)$).

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### 16. Augmented Encoder Destination Partition Contains Undefined Superscript N'

**Status**: [Pending]

**Quote**:
> then there is an equivalent trellis code $\mathbb{C}(G^N / \phi^\mu G^{N'}; C')$ based on the partition $G^N / \phi^\mu G^N$, where $C'$ is a $2^r$-state, rate-$[k + k(\Lambda')] / N\mu$ convolutional code, and $N$ is the dimension of $\Lambda$ or $\Lambda'$ as complex lattices.

**Feedback**:
The notation G^{N'} introduces an undefined symbol N'. The proof text itself correctly writes G^N/φ^μ G^N throughout, and the same typo recurs in the alternative real form as Z^{N'}. Since the partition involves the same lattice dimension N throughout, rewrite 'G^{N'}' as 'G^N' and 'Z^{N'}' as 'Z^N' in both the complex and real forms of Lemma 5.

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### 17. Regularity Condition (b) Contains Wrong φ Exponent in Partition Chain Equation

**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 complex-lattice chain corresponding to Z²/RZ²/2Z²/2RZ² is G/φG/φ²G/φ³G, since φ²G = 2iG corresponds to 2Z² and φ³G corresponds to 2RZ². The equation as written shows φ³G twice (at positions 3 and 4), omitting φ²G entirely. Rewrite 'G/φG/φ³G/φ³G' as 'G/φG/φ²G/φ³G.'

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### 18. Coset Representatives Claim Conflates Full Partition with Individual Two-Way Sub-Partition

**Status**: [Pending]

**Quote**:
> Thus the $2^{K}$ binary linear combinations $\{\sum a_{k}\mathbf{g}_{k}\}$ of the generators $\mathbf{g}_{k}$ are a system of coset representatives $[\Lambda_{k} / \Lambda_{k + 1}]$ for the partition $\Lambda_{k} / \Lambda_{k + 1}$ (this is a special case of a general result for groups of order $2^{K}$ — binary groups—discussed in part II).

**Feedback**:
The 2^K binary linear combinations {Σ a_k g_k} form a system of coset representatives for the full partition Λ/Λ' (order 2^K), not for the individual two-way partition Λ_k/Λ_{k+1} (order 2, needing only {0, g_k}). The bracket notation [Λ_k/Λ_{k+1}] denotes a 2-element set, not a 2^K-element set. Rewrite '[Λ_k/Λ_{k+1}] for the partition Λ_k/Λ_{k+1}' as '[Λ/Λ'] for the full partition Λ/Λ'.'

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### 19. Approximate Equality Symbol Used for Exact Lattice Equality in Lemma 2

**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 and each partition $\Lambda_k / \Lambda_{k+1}$ is two-way, $0 \leq k \leq K-1$.

**Feedback**:
The proof sketch confirms 'Λ_K = Λ' is certainly such a lattice,' requiring exact equality. Using ≈ instead of = introduces an ambiguity: if Λ_K ≠ Λ' exactly, the coset decomposition Λ = Λ' + {Σ a_k g_k} stated immediately after would not follow. Rewrite 'Λ_K ≈ Λ'' as 'Λ_K = Λ'.'

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### 20. State Count Parameter ν vs. 2^r Conflation in Trellis Description

**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 $C$.

**Feedback**:
The preceding paragraph defines the number of encoder states as 2^ν where ν is the constraint length. For the four-state example, r=1 (rate 1/2) but ν=2 (two memory elements), giving 2^ν=4 states ≠ 2^r=2. The label '2^r-state' is therefore inconsistent with the definition 2^ν states given just above. The paper uses '2^r-state' as shorthand throughout, but the identification ν=r is not stated and is not generally valid. It would be helpful to rewrite '2^{r}-state' as '2^{\nu}-state' (or clarify that ν=r is assumed for the codes considered here).

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### 21. Fundamental Coding Gain Formula γ = 2^{−e}d²_min Contradicts General Definition

**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 general definition from Section IV-B is γ = 2^{−ρ}d²_min where ρ is the normalized redundancy per two dimensions. For the 8-state one-dimensional code, ρ=2 and d²_min=10, giving γ=10/4=2.5, which matches the table. But the formula γ=2^{−e}d²_min with e=3 (state count 2^3=8) gives 2^{−3}·10=1.25, which does not match. The formula γ=2^{−e}d²_min is therefore not generally correct; it only coincidentally matches the 4-state one-dimensional entry where e=ρ=2. Rewrite 'The fundamental coding gain $\gamma$ is given by the formula $2^{-e}d_{\mathrm{min}}^2$' as 'The fundamental coding gain $\gamma$ is given by the formula $2^{-\rho}d_{\mathrm{min}}^2$.'

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### 22. GCS-Type Code Table VI: Λ' Listed as 2Z^8 Instead of 2Z^4

**Status**: [Pending]

**Quote**:
> |  4 | Z^{4} | 2Z^{8} | 8 | 3/4 | 1/2 | 4 | 2^{3/2} | 4.52 | 44 | 64 | 3.82  |

**Feedback**:
The section text describes this code as based on the four-dimensional partition Z^4/2Z^4 (N=4). The sublattice Λ' must have the same ambient dimension as Λ=Z^4, so Λ'=2Z^4, not 2Z^8 (an 8-dimensional lattice). The first Wei-code entry in Table VII confirms this: the 32-state code is listed as Z^4/2Z^4 and is explicitly identified as a GCS-type code. The '8' superscript in '2Z^8' appears to be a typographical error for the exponent on Z. Rewrite '2Z^{8}' as '2Z^{4}.'

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### 23. Note Added in Proof: D₄/RD₄ vs. D₄/2D₄ Partition Label Discrepancy Creates Concrete Inconsistency

**Status**: [Pending]

**Quote**:
> Note added in proof: J. Chow (private communication) has obtained values of $\tilde{N}_0 = 88$ and $\gamma_{\mathrm{eff}} = 3.88$ dB for the 16-state $D_4 / RD_4$ code, and of $d_{\min}^2 = 6$, $\tilde{N}_0 = 16$, and $\gamma_{\mathrm{eff}} = 4.37$ dB for the 64-state $D_4 / RD_4$ code.

**Feedback**:
The table body lists 4-dimensional codes under the partition D₄/2D₄, but the note refers to D₄/RD₄ codes. These are different partitions: |D₄/2D₄|=16 while |D₄/RD₄|=2. More concretely, the note gives d²_min=6 for the 64-state D₄/RD₄ code, while the table shows d²_min=8 for the 64-state D₄/2D₄ code—a concrete numerical discrepancy. It is not clear whether the note intends to correct the table's partition label or to report results for a separate code family. Rewrite the note to specify whether D₄/RD₄ refers to the same codes as the D₄/2D₄ rows or to a distinct family, and if the same, correct d²_min=8 to d²_min=6 and recompute γ and γ_eff accordingly.

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### 24. PSK Minimum Squared Distances List Nearest-Neighbor Distances Rather Than Intra-Coset Distances

**Status**: [Pending]

**Quote**:
> Although the minimum squared distances within these constellations are somewhat different from those in lattice partitions (e.g., 0.152/0.586/2/4/∞ for these constellations, compared to 1/2/4/8/16 for the comparable two-dimensional lattice partition $Z^2 / RZ^2 / 2Z^2 / 2RZ^2 / 4Z^2$)

**Feedback**:
The values 0.152/0.586/2/4/∞ are the nearest-neighbor distances of each constellation in the chain (16PSK, 8PSK, 4PSK, 2PSK, 1PSK), not the intra-coset distances of each partition step. For the partition 16PSK/8PSK, the minimum squared distance between two 16PSK points in the same 8PSK coset is 2 − 2cos(2π·2/16) ≈ 0.586, not 0.152. The comparison to lattice partition distances 1/2/4/8/16 is therefore not like-for-like. It would be helpful to clarify whether the listed values are nearest-neighbor distances of each constellation or intra-coset distances of each partition step, and if the latter, to recompute them accordingly.

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### 25. Effective Coding Gain for Z^4/2D₄ Entry: Extremely Large Error Coefficient Warrants Explicit Verification

**Status**: [Pending]

**Quote**:
> |  4 | Z^{4} | 2D_{4} | 128 | 4/5 | 1/2 | 6 | 6/2^{1/2} | 6.28 | 728 | 2040 | 4.77  |

**Feedback**:
The Ñ₀=728 value for the Z^4/2D₄ 128-state code is strikingly large—larger than the kissing number of E₈ (240) and comparable to that of Λ₁₆ (4320/8=540 per two dimensions). Using the paper's rule of thumb with reference Ñ₀=4: 728/4=182≈2^{7.51}, penalty≈7.51×0.2=1.50 dB, giving γ_eff≈6.28−1.50=4.78 dB, consistent with the tabulated 4.77 dB. However, the paper never states the reference Ñ₀=4 explicitly in the table caption, and the magnitude of Ñ₀=728 raises the question of whether the nearest-neighbor multiplicity for the Z^4/2D₄ partition has been computed correctly. It would be helpful to verify this value and to state the reference Ñ₀ used in the penalty calculation explicitly in the table caption so readers can reproduce the γ_eff entries.

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