# 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.

**Eight-Class Taxonomy Is Presented as Exhaustive Without Formal Justification**

Section VI introduces eight code classes (I–VIII) defined by three binary structural choices—partition type (single vs. two-level chain), encoder rate (k/2k vs. k/(k+1)), and constraint length (k vs. 2k)—and Section VII draws broad performance conclusions from this taxonomy, including the folk theorem that 'it takes two states to get 1.5 dB, four states to get 3 dB…' However, the paper never establishes that these three binary axes are the only relevant structural dimensions, nor that the resulting eight classes cover all coset codes based on binary lattice partitions. Codes with non-uniform memory distributions, rate-k/(k+r) encoders with r > 1, or partition chains of depth greater than two are excluded without justification. The discussion in Section VII acknowledges that 'there are many ways to modulate,' which implicitly concedes the classification may be illustrative rather than complete. It would be helpful to either state explicitly that the taxonomy is a representative sample rather than a provably complete enumeration, or to provide a formal argument—perhaps via Lemma 5's equivalence to G^N/φ^μG^N codes—showing that no structurally distinct class is omitted. As stated, readers may incorrectly infer that the eight classes exhaust all possible binary-lattice trellis code constructions, lending unwarranted authority to the Section VII design guidelines.

**Effective Coding Gain Rule of Thumb Is Applied Without Derivation or Stated Validity Range**

Throughout Sections V–VII, the paper converts fundamental coding gain γ and nearest-neighbor error coefficient Ñ₀ into an effective coding gain γ_eff using the rule that '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 is stated without derivation, without a reference, and without specifying the SNR range or union-bound conditions under which it is valid. The paper itself notes in Section V that 'in principle, the error coefficients at every distance ought to be considered,' yet γ_eff values in Tables IV–XI and Figure 12 are computed using only Ñ₀. For codes with very large nearest-neighbor multiplicities—such as the Class VI Λ_16/RΛ_16 code with Ñ₀ = 540, or the 128-state Z²/2RZ² Ungerboeck code with Ñ₁ = 624 and Ñ₂ = 2568—sub-dominant distance terms are likely to dominate at practical SNRs, making the single-term approximation potentially misleading. The conclusion in Section VII-2 that 'trellis codes are better than lattice codes' depends critically on this rule of thumb. It would be helpful to either provide a union-bound derivation justifying the 0.2 dB/factor-of-two rule, flag explicitly which γ_eff entries are unreliable because higher-order error coefficients dominate, or supplement with a transfer-function bound for at least one representative high-multiplicity code.

**Linearity and Distance-Invariance Conditions for Non-Mod-2 Partitions Are Incompletely Verified**

Lemma 6 (Section IV-C) establishes linearity of trellis codes when Λ' is a mod-2 lattice and the labeling is linear modulo Λ'. However, many codes in Tables VII–XI use partitions where Λ' is a mod-4 or deeper lattice—for example, E₈/RE₈ (depth 3), Λ_16/RΛ_16 (depth 4), and H_16/Λ_16 (depth 3). For these cases the paper asserts distance-invariance via the regularity condition (Section IV-C), but the verification that the labeling is regular—specifically that the minimum norm in coset Λ' + c(a) − c(a') equals the minimum norm in Λ' + c(a ⊕ a')—is only sketched for special cases. The group homomorphism property required for the coset structure to be preserved under complex Gaussian integer multiplication is not verified for the non-mod-2 cases, and the paper does not systematically identify which entries in Table XI are linear and which rely solely on regularity. Since the error coefficient calculations in Table XI depend on distance-invariance holding, it would be helpful to either extend Lemma 6 to cover mod-4 and higher-depth partitions with an explicit group-homomorphism argument, add a column to Table XI indicating linearity status, or add a remark identifying which entries rely on unverified regularity.

**Coding Gain Formula and Comparison Baseline Are Not Uniformly Verified Across All Redundancy Levels and Dimensionalities**

The fundamental coding gain γ(C) = 2^{−ρ(C)} d²_min(C) is defined in Section I-B and used throughout as the primary performance metric. In Section IV-B the same symbol is derived via two routes—one relative to Λ' and one relative to Λ—and the intermediate expressions implicitly assume that the reference uncoded system uses a constellation of the same spectral efficiency. This assumption is stated informally but never verified for codes where the underlying lattice Λ has nonzero redundancy ρ(Λ) > 0, such as the CS-type codes based on E₈/RE₈ (Table IX, ρ = 5/4), where the uncoded reference is not Z² but a denser lattice and the power baseline shifts. Additionally, when comparing codes of different dimensionality N in Tables VI–XI and Figure 12, the constellation expansion factors differ by up to 2^{1.75} ≈ 3.4×, and the approximation V(C) ≈ 2^{r(C)}V(Λ) used in Section IV-B may introduce non-negligible errors at finite constellation sizes. It would be helpful to add a brief lemma in Section IV-B fixing the spectral efficiency of the reference system and confirming the formula's validity for all ρ(Λ) > 0, and to include an annotation or secondary axis in Figure 12 grouping codes by ρ value so that readers can assess whether apparent gains from multidimensional codes survive the constellation expansion penalty in practice.

**Dual Lattice Definition Inconsistency Between Complex and Real Conventions Creates Ambiguity in Table I and Partition Chain Results**

Section II-H introduces two definitions of the dual lattice: a complex definition using the Hermitian inner product modulo φ^μ, and an alternative real definition using the real inner product modulo 2^m. The paper notes that these coincide when the φ-depth μ is even but differ by an R-rotation when μ is odd. This discrepancy is consequential because the dual partition chains used in Section III (e.g., Λ(r,n)^⊥ = Λ(n−r−1, n)) and the code formulas populating Table I are derived under the complex convention, while Table I lists real code formulas. For odd-depth lattices such as D₄ (depth 1), the self-duality claim in Section II-H uses the complex convention, yet the real code formula RD₄ = 2Z⁴ + (4,1,4) cited as the real dual is a rotated version—making the self-duality statement potentially confusing. It is not always clear which convention is in use when these dual relationships are invoked for partition chain properties in Figure 9 and Table III. It would be helpful to add a footnote to Table I clarifying which dual convention applies to each entry, and to add a clarifying remark immediately after the self-duality claim for D₄ specifying which inner product definition is being used.

**Minimum Squared Distance Formula for Indecomposable Lattices Lacks Explicit Verification**

The minimum squared distance formula for decomposable mod-4 real lattices (Section II-G) and its complex analogue in Lemma 4 assume that the minimum norm in each coset of 4Z^N is achieved by the coset representative itself, which holds when generators are binary (0/1-valued). The paper explicitly flags indecomposability for the Leech lattice Λ_24 and its half-lattice H_24 using prime notation (e.g., (12,5,8)' and (12,1,16)'), but does not re-derive the minimum distance formula under the modified generator structure that applies in the indecomposable case, nor does it verify that the values d²_min = 16 and d²_min = 8 listed in Table II for Λ_24 and H_24 follow from the same formula or from a separate argument. Similarly, the error coefficient formula for Class V codes—N₀ = N_{Λ'} + (2^{k+1} − k − 2)N²_Λ—is derived for a specific encoder T but then applied to all noncatastrophic rate-k/(k+1) encoders satisfying properties (a) and (b), without verifying that the path count is invariant across all such encoders. Since these lattice parameters and error coefficient formulas feed directly into the coding gain comparisons of Section VII, it would be helpful to either provide brief proof sketches for the indecomposable cases, cite specific results from Conway and Sloane that confirm the distance values, and restrict the N₀ formula explicitly to encoder T or prove its generality.

**Coding Gain Formula Assumes Distance-Invariance, Which Is Asserted Rather Than Verified for Non-Linear Codes**

The fundamental coding gain γ(C) = 2^{−ρ(C)} d²_min(C) is derived under the assumption that the code is distance-invariant (Section IV-B), meaning the distance distribution from any given code sequence to all others is the same regardless of which sequence is chosen as the reference. The paper acknowledges this assumption explicitly and states 'all codes in this paper are distance-invariant,' attributing this to the regularity of the labeling (Section IV-C). However, for non-linear trellis codes—those based on mod-4 or deeper partitions where the labeling is not linear modulo Λ′—the paper appeals to Ungerboeck-type regular labelings and condition (b) of the regularity criteria (that the Ungerboeck distance bound holds with equality). It would be helpful to verify this condition explicitly for each partition family used in Tables VII–XI, particularly for the mod-4 partitions such as Z²/4Z², D₄/2D₄, and E₈/RE₈, since the paper notes (Section IV-C) that condition (b) holds for Z²/RZ²/2Z²/2RZ² but does not provide an analogous verification for deeper chains. If distance-invariance fails for any of these cases, the reported d²_min and N₀ values—and consequently the effective coding gain γ_eff figures in Tables IX and X—may not characterize performance uniformly across all starting sequences.

**Geometrical Coding Gain Is Used as the Primary Performance Metric Without Accounting for the Divergence of Error Coefficients at High Gain**

The paper defines the fundamental coding gain γ(C) purely from d²_min and the fundamental volume V(C), and uses a rule of thumb that every factor-of-two increase in the normalized error coefficient Ñ₀ costs approximately 0.2 dB to compute an effective coding gain γ_eff. This rule of thumb is acknowledged to be approximate and valid only 'at error rates of the order of 10⁻⁶.' Readers might note that for several high-gain codes in Tables VII, VIII, and XI—particularly the Class V codes based on H₁₆/Λ₁₆ (Ñ₀ not reported) and the 16-state Λ₁₆/RΛ₁₆ Class I code—the error coefficients are either unknown or described as 'very large,' yet the fundamental coding gains of 6–7 dB are still reported and compared. The paper itself cautions that 'the very large error coefficients of lattice codes will mean that their effective performance will be significantly inferior' (Section VII, point 2), but applies the same caveat inconsistently: γ_eff is omitted for lattice codes due to large Ñ₀, yet γ_eff is reported for Class V trellis codes with similarly large or unknown Ñ₀. It would be helpful to apply a consistent criterion—either reporting γ_eff for all codes with known Ñ₀ or flagging all codes with Ñ₀ above a stated threshold—so that the performance comparisons in Figure 12 and Table XI are not implicitly biased toward codes whose error coefficients happen to be known.

**Status**: [Pending]

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## Detailed Comments (25)

### 1. Shape Gain Formula: Variable Name Collision Between n = N/2 and Information-Rate n

**Status**: [Pending]

**Quote**:
> the shape gain of an $N$-sphere over an $N$-cube, which for $N$ even is $G_{\otimes} = \pi (n + 1) / [6(n!)^{1 / n}]$, where $n = N / 2$; thus $G_{\otimes} = \pi /3$ (0.20 dB) for $N = 2$, $\pi /2^{3 / 2}$ (0.46 dB) for $N = 4$, $5\pi /6(24)^{1 / 4}$ (0.73 dB) for $N = 8$, with a limit of $\pi e / 6$ (1.53 dB) as $N\to \infty$ [1].)

**Feedback**:
The local variable n = N/2 introduced in this parenthetical collides with the information-rate variable n used earlier in the same section ('To transmit n bits per N dimensions'). A reader carrying the earlier definition into the shape-gain formula will misread it entirely. The numerical values are correct (verified: N=2 gives π(2)/[6·1]=π/3 ✓; N=4 gives π(3)/[6·√2]=π/2^{3/2} ✓; N=8 gives 5π/[6·24^{1/4}] ✓), so the formula itself is fine. It would be helpful to rewrite the parenthetical using a distinct symbol, e.g., 'where m = N/2,' to avoid the collision with the information-rate variable n.

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### 2. Constellation Expansion Power Cost Presented as Exact Rather Than Asymptotic

**Status**: [Pending]

**Quote**:
> ,  $n - k$  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 claim that a constellation expansion factor of 2^{ρ(C)} translates into an exact average power cost of 2^{ρ(C)} holds only asymptotically for large constellations with a fixed boundary shape. For finite constellations, the power cost depends on the specific boundary and lattice geometry. The paper acknowledges finite-constellation effects through γ_s in the very next paragraph, yet presents the 2^{ρ(C)} power cost here as exact. Rewriting 'This translates into an average power cost of a factor of $2^{\rho(C)}$' as 'This translates into an asymptotic average power cost of a factor of $2^{\rho(C)}$ (exact for large constellations with a fixed boundary shape)' would make the approximation explicit and consistent with the surrounding discussion.

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### 3. Claimed 6 dB Gain at 128 States Contradicts Section VII Folk Theorem Placing 6 dB at 256 States

**Status**: [Pending]

**Quote**:
> Through a technique called "mapping by set partitioning" of one- and two-dimensional signal constellations, combined with binary convolutional codes, Ungerboeck showed how coding gains of the order of 3 dB could be obtained with simple four-state codes, while gains of up to 6 dB could be obtained with more complex (128-state) codes.

**Feedback**:
Section VII's folk theorem states '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.' This places the 6 dB milestone at 256 states, not 128. The 128-state Ungerboeck code achieves a fundamental coding gain of 6.02 dB, but the effective coding gain after error-coefficient penalty is lower (Table V shows ~4.74–5.01 dB for 128-state codes). The introduction's claim of 'up to 6 dB' at 128 states is therefore in tension with the paper's own Section VII analysis. Readers might note that rewriting '(128-state) codes' as '(128- to 256-state) codes' and adding a forward reference to Section VII's folk theorem would resolve this internal inconsistency.

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### 4. R Matrix Is a Scaled Improper Rotation (Reflection), Not a Pure Rotation

**Status**: [Pending]

**Quote**:
> $R\mathbb{Z}^2$ is a version of $\mathbb{Z}^2$ obtained by rotating $\mathbb{Z}^2$ by $45^\circ$ and scaling by $2^{1/2}$

**Feedback**:
The matrix R = [[1,1],[1,−1]] has determinant det(R) = (1)(−1) − (1)(1) = −2, so it is a scaled improper orthogonal transformation (a reflection composed with a 45° rotation and √2 scaling), not a pure rotation. A true rotation by 45° scaled by √2 would be [[1,−1],[1,1]] with determinant +2. The distinction matters when the paper later uses R in the context of 'orthogonal transformation' and when discussing whether RΛ is a rotated or reflected version of Λ. It would be helpful to rewrite 'obtained by rotating Z² by 45° and scaling by 2^{1/2}' as 'obtained by a scaled improper orthogonal transformation (reflection composed with a 45° rotation, scaling by 2^{1/2})' to accurately describe the geometric action.

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### 5. Cartesian Product Density Comparison: Second Lattice Symbol Is a Duplicate Typo

**Status**: [Pending]

**Quote**:
> $\Lambda_N^M$ 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.

**Feedback**:
The sentence compares 'Λ_N^M' to 'Λ_N^M'—the same expression appears on both sides of 'than does,' which is a typographical error. From context and the surrounding argument, the intended comparison is between the M-fold Cartesian product of the N-dimensional lattice Λ_N (written Λ_N^M) and the N-fold Cartesian product of the M-dimensional lattice Λ_M (written Λ_M^N), both living in MN-dimensional space. Verification: γ(Λ_N^M) = γ(Λ_N) and γ(Λ_M^N) = γ(Λ_M), so Λ_N^M is denser iff γ(Λ_N) > γ(Λ_M). Rewrite 'than does Λ_N^M' as 'than does Λ_M^N.'

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### 6. Coset Decomposition Chain Order Inconsistent Between Abstract Formula and Binary 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 $[\Lambda / \Lambda']$.

**Feedback**:
The abstract formula places the finer-partition representative [Λ'/Λ''] second and the coarser [Λ/Λ'] last, while the concrete binary example immediately following writes Z = [Z/2Z] + [2Z/4Z] + … with the coarser partition [Z/2Z] (i.e., a₀) appearing first. Mapping the binary example onto the abstract formula (Λ=Z, Λ'=2Z, Λ''=4Z) gives [Λ/Λ'] = [Z/2Z] and [Λ'/Λ''] = [2Z/4Z], so the abstract formula places [Z/2Z] last while the binary expansion places it first. Since addition is commutative the equation is algebraically correct, but the ordering convention is inconsistent between the abstract formula and the concrete example. It would be helpful to add an explicit note that addition is commutative so the order is immaterial, or to rewrite the abstract formula to match the binary-example ordering.

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### 7. Subcode Inequality K − K' ≤ K' Attributed to Wrong 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 of C (dimension K) only implies K − K' ≤ K, not K − K' ≤ K'. To obtain K − K' ≤ K', one needs C' to be a subcode of C₁ (the code associated with Λ_e, which has dimension K'). Rewrite 'a subcode of the code C, so K − K' ≤ K'' as 'a subcode of the code C₁ associated with Λ_e, which has dimension K', so K − K' ≤ K'' because the bound follows from the dimension of C₁, not C.

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### 8. Regularity Condition (b) Cites Incorrect Exponents in Partition Chain

**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 real chain Z²/RZ²/2Z²/2RZ² corresponds to the complex chain G/φG/φ²G/φ³G (exponents 0,1,2,3), since φ⁰G=Z², φ¹G=RZ², φ²G=2Z², φ³G=2RZ². The paper writes G/φG/φ³G/φ³G, which has the third and fourth elements identical (both φ³G), implying a trivial last step of index 1 and omitting φ²G=2Z² entirely. This is a typographical error in the exponents. Rewrite 'G / \phi G / \phi^3 G / \phi^3 G' as 'G / \phi G / \phi^2 G / \phi^3 G.'

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### 9. Dual Lattice Verification Sentence Uses 'n' and 'u' Instead of Depth Parameter 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 $\Lambda / \phi^n \pmb{G}^N$, and that the dimensions are such that the informativity of $\Lambda^\perp$ is equal to the redundancy of $\Lambda$ and vice versa.

**Feedback**:
The section consistently uses μ as the depth parameter (e.g., 'φ^μ G^N,' 'φ^{μ−1} C_{μ−1}'), yet this verification sentence switches to 'n' in 'C_0^⊥/…/C_{n−1}^⊥,' 'Λ^⊥/φ^n G^N,' and 'Λ/φ^n G^N,' and to 'u' in 'mod φ^u.' The chain should have μ terms (indices 0 through μ−1), and the quotient lattices should reference φ^μ G^N. The dual code formula displayed just above uses φ^μ G^N correctly, making the switch to 'n' and 'u' an inconsistency. Rewrite 'C_0^⊥/…/C_{n−1}^⊥ … Λ^⊥/φ^n G^N … mod φ^u … Λ/φ^n G^N' as 'C_0^⊥/…/C_{μ−1}^⊥ … Λ^⊥/φ^μ G^N … mod φ^μ … Λ/φ^μ G^N.'

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### 10. Primal Code Formula Uses 'φ^n G^N' Instead of 'φ^μ G^N' for Depth-μ Lattice

**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**:
For a lattice of depth μ, the leading term of the code formula should be φ^μ G^N, representing the sublattice of all vectors divisible by φ^μ. The formula as displayed uses φ^n G^N (with 'n' rather than 'μ'), inconsistent with the rest of the section and with the dual formula immediately following, which correctly writes φ^μ G^N. This inconsistency propagates into the verification sentence (Comment 9 above). Rewrite 'Λ = φ^n G^N + φ^{μ−1} C_{μ−1} + … + C_0' as 'Λ = φ^μ G^N + φ^{μ−1} C_{μ−1} + … + C_0.'

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### 11. D₄ Self-Duality Claim Requires Clarification of Which Dual Convention Is Used

**Status**: [Pending]

**Quote**:
> ts dual under this alternative definition is still itself, because the $(8,4,4)$ code is self-dual. The Schlafli lattice $D_{4}$ has depth 1 and has real code formula $D_{4} = 2Z^{4} + (4,3,2)$; its dual under this alternative definition is $RD_{4}$, which has code formula $RD_{4} = 2Z^{4} + (4,1,4)$, since the $(4,3,2)$ and $(4,1,4)$ codes are duals.

**Feedback**:
Earlier in Section II-H the paper establishes D₄ is self-dual under the complex Hermitian convention. Here the same section states that under the real inner product convention the dual of D₄ is RD₄ (not D₄ itself), because D₄ has odd φ-depth μ=1. These two statements are not contradictory but they use different conventions, and the text does not flag the switch. A reader who does not notice the convention change will conclude that D₄ is simultaneously self-dual and not self-dual. It would be helpful to add the phrase 'under the complex inner product definition' when asserting D₄ is self-dual, and 'under the real inner product definition' when asserting its dual is RD₄, to prevent conflation of the two conventions.

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### 12. Complex Code Formula for H₁₆ in Table I Lists G^4 Instead of G^8

**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 Λ(1,3) = H₁₆, the general formula Λ(r,n) = φ^{n−r} G^N + … with N = 2^n = 2^3 = 8 gives Λ(1,3) = φ² G^8 + φ RM(2,3) + RM(1,3) = φ² G^8 + φ(8,7,2) + (8,4,4). H₁₆ is a 16-dimensional real lattice (2N=16, so N=8 complex dimensions), requiring base lattice G^8. The table entry lists φ²G^4, which is the base lattice for E₈ = Λ(0,2) (N=2^2=4). The G^4 appears to be a typographical error copied from the E₈ row. Rewrite 'φ^{2}G^{4} + φ(8,7,2) + (8,4,4)' as 'φ^{2}G^{8} + φ(8,7,2) + (8,4,4).'

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### 13. Depth Subscript Typo: 'D_k' Should Be 'D_4' in Depth Example

**Status**: [Pending]

**Quote**:
> For example, the depth of $D_{k}$ is 1, its redundancy and informativity are both equal to 1, and its normalized redundancy and informativity are both equal to $1/2$.

**Feedback**:
The running example throughout Section II-E is D₄ (the Schläfli lattice), and the numerical values given (depth 1, redundancy 1, informativity 1, normalized values 1/2) are verified for D₄: r(D₄) = log₂|Z⁴/D₄| = log₂(2) = 1, k(D₄) = N·μ − r = 2·1 − 1 = 1, ρ = κ = 1/2. The subscript k is used elsewhere as a code rate parameter, making 'D_k' ambiguous. Rewrite 'the depth of $D_{k}$' as 'the depth of $D_{4}$.'

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### 14. Lemma 2 Statement Uses Approximate Equality '≈' Where Exact Equality Is Required

**Status**: [Pending]

**Quote**:
> **Lemma 2**: Let $\Lambda$ and $\Lambda'$ be binary lattices such that $\Lambda'$ is a sublattice of $\Lambda$, and let $|\Lambda / \Lambda'| = 2^K$. 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 on the very next line writes 'Λ_K = Λ'' (exact equality), confirming the intent is exact equality at the terminal lattice. Using '≈' in the lemma statement introduces an ambiguity: does the chain end at Λ' exactly, or only approximately? This is likely an OCR artifact for '='. Rewrite '$\Lambda_K \approx \Lambda'$' as '$\Lambda_K = \Lambda'$' in the Lemma 2 statement.

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### 15. Coset Representatives Notation Conflates Full Partition with Individual Two-Way 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 $\Lambda_{k} / \Lambda_{k + 1}$

**Feedback**:
Each individual two-way partition Λ_k/Λ_{k+1} has only 2 cosets, represented by {0, g_k}, not 2^K combinations. The 2^K binary linear combinations {Σ a_k g_k} are coset representatives for the full partition Λ/Λ', not for each individual Λ_k/Λ_{k+1}. Rewrite 'are a system of coset representatives [Λ_k/Λ_{k+1}] for the partition Λ_k/Λ_{k+1}' as 'are a system of coset representatives for the full partition Λ/Λ', where for each k the two vectors {a_k g_k, a_k ∈ {0,1}} are coset representatives for Λ_k/Λ_{k+1}.'

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### 16. Incorrect Intermediate Term in Volume Chain for V(C)

**Status**: [Pending]

**Quote**:
> 2k(C)/N$ and $\rho(C) = 2r(C)/N$, respectively. Since $V(\mathbb{C}) = V(\Lambda_0) = 2^{-k(C)}V(\Lambda') = 2^{k+r}V(\Lambda') = 2^{r(C)}V(\Lambda)$, w

**Feedback**:
The second equality in this chain reads 2^{−k}V(Λ') = 2^{k+r}V(Λ'), which would require 2^{−k} = 2^{k+r}, i.e., k = −(k+r)/2—clearly false for positive k, r. The intended step uses V(Λ') = 2^{k+r}V(Λ) (from Lemma 1) to write V(Λ₀) = 2^{−k}V(Λ') = 2^{−k}·2^{k+r}V(Λ) = 2^r V(Λ). The spurious middle term '2^{k+r}V(Λ')' should be removed. Rewrite the chain as 'V(C) = V(Λ₀) = 2^{−k(C)}V(Λ') = 2^{r(C)}V(Λ).'

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### 17. Alternative Form of Lemma 5: Rate Formula Has Factor-of-2 Ambiguity for Odd-Depth Case

**Status**: [Pending]

**Quote**:
> An alternative form of Lemma 5 is as follows. If the 2-depth of $\Lambda'$ is $m$, then $\mathbb{C}(\Lambda / \Lambda'; C)$ is equivalent to a code $\mathbb{C}(Z^N / 2^m Z^{N'}; C')$ based on the partition $Z^N / 2^m Z^N$, where $C'$ is a $2^r$-state, rate- $[Nm - r - r(\Lambda)] / Nm$ convolutional code, and $N$ is the dimension of $\Lambda$ or $\Lambda'$ as real lattices.

**Feedback**:
For the odd-depth case μ = 2m−1, the real partition Z^{2N}/2^m Z^{2N} has 2^{2Nm} cosets, so the augmented encoder outputs 2Nm bits per step. The information count is k + k(Λ') + N (the extra N uncoded bits from the φ^μ G^N/φ^{μ+1}G^N extension), which equals 2Nm − r − r(Λ). This gives rate [2Nm − r − r(Λ)]/(2Nm). The stated formula [Nm − r − r(Λ)]/(Nm) matches only if N denotes the complex dimension (half the real dimension), but the text says 'N is the dimension of Λ or Λ' as real lattices.' This introduces a factor-of-2 discrepancy in the denominator for the odd-depth case. It would be helpful to rewrite the rate as '[2Nm − r − r(Λ)]/(2Nm)' (with N as the real dimension) or clarify that N here denotes the complex dimension.

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### 18. Minimum Squared Distance Formula in Lemma 6 Proof Sketch Is Incomplete

**Status**: [Pending]

**Quote**:
> The minimum squared distance between code sequences corresponding to distinct codewords of $C'$ is $d_{H}(C')$, and $d_{\min}^{2}(2Z^{N}) = 4$.

**Feedback**:
The proof sketch states the minimum squared distance between distinct-codeword sequences is d_H(C'), and separately that d²_min(2Z^N) = 4, but does not explain why the overall minimum is min[4, d_H(C')] as stated in Lemma 6. The two contributions come from different sources: (i) cross-coset differences contribute at least d_H(C') via the Ungerboeck bound, and (ii) within-coset differences (parallel transitions) can be as small as d²_min(2Z^N) = 4. The proof sketch omits case (ii) entirely. It would be helpful to add after 'd²_min(2Z^N) = 4' the sentence: 'Sequences sharing the same coset-label sequence but differing within a coset of 2Z^N can be as close as d²_min(2Z^N) = 4; hence the overall minimum is min[4, d_H(C')].'

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### 19. Fundamental Coding Gain Formula Stated as '2^{−e} d²_min' Instead of '2^{−ρ} d²_min'

**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 the fundamental coding gain as γ = 2^{−ρ} · d²_min, where ρ is the normalized redundancy per two dimensions. For the one-dimensional Ungerboeck codes, ρ = 2 and the state-count exponent e = 2 (four states), so the formula 2^{−e} d²_min happens to give the correct numerical answer. But for the two-dimensional codes with ρ = 1 and e = 3 (eight states), 2^{−e} d²_min = 2^{−3}·5 = 0.625, not the correct γ = 2^{−1}·5 = 2.5. The formula should read 2^{−ρ} d²_min. Rewrite 'The fundamental coding gain γ is given by the formula $2^{-e}d_{\mathrm{min}}^2$' as 'The fundamental coding gain γ is given by the formula $2^{-\rho}d_{\mathrm{min}}^2$.'

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### 20. Error Coefficient Formula for Class V Codes: Generality Across Encoders Not Established

**Status**: [Pending]

**Quote**:
> The multiplicity $N_0$ of sequences at distance $d_{\min}^2(\Lambda')$ from any given sequence starting at any given time is
> 
> $$
> N _ {0} = N _ {\Lambda^ {\prime}} + (2 ^ {k + 1} - k - 2) N _ {\Lambda} ^ {2}
> $$

**Feedback**:
The coefficient (2^{k+1} − k − 2) is derived by counting nonzero paths of lengths 2 through k+1 in the trellis: 2^k−1 paths of length 2, 2^{k−1}−1 of length 3, …, 1 of length k+1, summing to (2^k−1)+(2^{k−1}−1)+…+1 = 2^{k+1}−k−2. This derivation is stated to hold 'for any noncatastrophic rate-k/(k+1) encoder' (citing Forney [26]), but the formula is derived for the specific encoder T described in the section. It is not verified that all noncatastrophic rate-k/(k+1) encoders satisfying properties (a) and (b) yield the same path count. Since this formula feeds directly into the γ_eff comparisons of Section VII, it would be helpful to either restrict the formula explicitly to encoder T or cite the specific result from [26] that establishes its generality.

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### 21. Duplicate Row for D₈/RE₈ in Table III with Inconsistent Orders and κ Values

**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 distinct rows for the partition D₈/RE₈, both 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, |D₈/RE₈| = V(RE₈)/V(D₈) is a fixed number, and κ(Λ/Λ') is determined by Λ' alone. Having two rows with the same lattice pair but different orders and κ values is a logical contradiction: one of these rows must be mislabeled and actually represents a different partition (e.g., Z₈/RE₈ or E₈/RE₈). Readers might note that the second D₈/RE₈ row should be checked against the partition chain in Fig. 9 and relabeled to correctly identify the lattice pair it represents.

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### 22. Partition Order for Z⁴/RD₄ in Table III Appears Inconsistent with Lemma 1

**Status**: [Pending]

**Quote**:
> |  Z4 | RD4 | 4 | 8 | 2 | 1/2 | 0 | 1 | 4 | 16 | 2  |

**Feedback**:
By Lemma 1, |Λ/Λ'| = V(Λ')/V(Λ). From Table II, V(Z⁴) = 1 and V(D₄) = 2 (since r(D₄) = 1). Since RD₄ is a rotation of D₄, V(RD₄) = V(D₄) = 2. Thus |Z⁴/RD₄| = V(RD₄)/V(Z⁴) = 2/1 = 2, not 8 as listed. The row above (Z⁴/D₄) correctly lists order 2. For the order to be 8, the sublattice would need to be a coarser lattice such as 2D₄ or a different lattice entirely. It would be helpful to rewrite the order entry for Z⁴/RD₄ from 8 to 2, or identify the correct sublattice that yields order 8 and relabel the row accordingly.

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### 23. Effective Coding Gain Rule of Thumb Applied Without Stating Reference Ñ₀ Baseline

**Status**: [Pending]

**Quote**:
> In this paper, we will use the rule of thumb that 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 will enable us to compute an effective coding gain $\gamma_{\mathrm{eff}}$ (in dB), normalized for the error coefficient $\tilde{N}_0$.

**Feedback**:
The rule of thumb is stated without specifying the reference value of Ñ₀ against which 'factor of two increases' are measured. Checking the GCS code (Table VI): γ = 4.52 dB, N₀ = 44, Ñ₀ = 44·(2/4) = 22. If the reference is Ñ₀ = 1, the penalty is 0.2·log₂(22) ≈ 0.89 dB, giving γ_eff ≈ 3.63 dB—but the table reports 3.82 dB. If the reference is Ñ₀ = 4 (the uncoded Z² kissing number normalized to 2D), the ratio is 22/4 = 5.5, penalty ≈ 0.2·log₂(5.5) ≈ 0.51 dB, giving γ_eff ≈ 4.01 dB—still not 3.82 dB. The exact baseline is not recoverable from the text alone. It would be helpful to add a sentence specifying the exact reference value of Ñ₀ used in the penalty formula so that all γ_eff entries in Tables IV–XI are reproducible.

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### 24. Folk Theorem State Counts Inconsistent with Class I/II/IV Code Parameters in Table XI

**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 folk theorem is presented as a general rule, but Table XI shows Class I codes achieving γ = 2^{3/2} (4.52 dB) with only 4 states (E₈/RE₈, Class I, 2^r = 4) and γ = 4 (6.02 dB) with only 16 states (Λ₁₆/RΛ₁₆, Class I, 2^r = 16). The folk theorem places these milestones at 16 and 256 states respectively. The paper itself acknowledges this in the surrounding text ('there are codes that achieve this fundamental coding gain with as few as eight states … but with very large error coefficients'), but the folk theorem as stated does not include the qualifier 'with reasonably small error coefficient' in the milestone numbers themselves—it adds it only at the end. Since the Class I codes with large Ñ₀ are explicitly listed in Table XI and compared in Figure 12, it would be helpful to rewrite the folk theorem to make the error-coefficient qualifier integral to each milestone, not a trailing clause.

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### 25. Inconsistent Application of γ_eff: Omitted for Lattice Codes but Reported for Class V Codes with Similarly Large Ñ₀

**Status**: [Pending]

**Quote**:
> We have not even given an effective coding gain $\gamma_{\mathrm{eff}}$ for lattice codes because our rule of thumb is questionable when the number of nearest neighbors is so high. Nonetheless, it seems clear that the very large error coefficients of lattice codes will mean that their effective performance will be significantly inferior to that of comparable trellis codes.

**Feedback**:
The paper withholds γ_eff for lattice codes due to large Ñ₀, yet Table XI reports γ_eff for Class V codes (e.g., the 128-state Λ₁₆/RΛ₁₆ code and the 64-state H₁₆/Λ₁₆ code) whose Ñ₀ entries are blank ('unknown but large') or whose error coefficients are described as 'outsize.' The conclusion in Section VII-2 that 'trellis codes are better than lattice codes' depends on this asymmetric treatment. It would be helpful to apply a consistent criterion—either reporting γ_eff for all codes with known Ñ₀ or flagging all codes with Ñ₀ above a stated threshold—so that the performance comparisons in Figure 12 and Table XI are not implicitly biased toward codes whose error coefficients happen to be known.

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