# Coset Codes—Part I: Introduction and Geometrical Classification

**Date**: 03/07/2026
**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.

**Alignment of Abstract Claims with Scope Limitations**

The Abstract suggests that practically all known good constructive coding techniques can be characterized as coset codes, while Section I-B restricts the analysis to lattice-type coset codes. Readers might note that this discrepancy creates ambiguity regarding the generality of the classification, particularly for non-lattice group structures. It would be helpful to align the Abstract with the body text by specifying the focus on lattice-type codes or expanding the analysis to address the broader claim.

**Rigor of Effective Coding Gain Estimation**

The paper utilizes a heuristic rule of thumb regarding error coefficients to calculate effective coding gain, which is used extensively in performance tables and plots. While acknowledged as an approximation, relying on this unverified metric for primary comparisons may obscure true trade-offs without a sensitivity analysis. It would be helpful to provide a more rigorous bound on how variations in error coefficients affect performance or explicitly label these values as heuristic estimates to maintain theoretical rigor.

**Treatment of Finite Constellation and Shape Gain**

The analysis focuses on fundamental coding gain derived from infinite lattice structures, excluding shape gain which can contribute significantly in practical finite block length systems. Readers might note that dismissing shape gain could understate the performance potential of codes amenable to spherical shaping. It would be helpful to include a discussion on how constellation boundaries interact with the presented code structures to provide a more complete picture of practical utility.

**Transparency of Comparative Data and Complexity Metrics**

Key comparative tables rely on unpublished work or private communications for critical parameters like error coefficients, and decoding complexity values depend on a companion paper. This reliance limits the reproducibility of the conclusions without access to external sources. It would be helpful to include specific derivations or reference published sources where these parameters are independently verified to ensure the comparative claims are fully substantiated.

**Consistency of Dimension and Lattice Definitions**

There are potential inconsistencies in the definition of the dimension parameter $N$ across sections, and the fundamental volume definition assumes a time-zero lattice structure that may not always hold. Readers might note that ambiguity in these definitions could lead to confusion in calculating coding gains or applying the framework to non-standard codes. It would be helpful to standardize the definition of $N$ and specify the conditions under which the time-zero lattice assumption is guaranteed to be valid.

**Clarification Needed on Decoding Complexity Formula**

It would be helpful to clarify the derivation of the decoding complexity values ($N_D$) in Table IV relative to the formula provided in Section V. The text states that complexity is $eta 2^{k+r}$ where $2^{k+r}$ is the number of branches per stage. However, for a trellis with $2^e$ states and rate $k/(k+r)$, the total number of branches per stage is typically $2^e 	imes 2^k$. For the 16-state code ($e=4, k=2, r=1$), the text formula yields $2^{3}=8$ branches, whereas the standard trellis structure implies $16 	imes 4 = 64$ branches. The table values (e.g., 120 for the 16-state code) align more closely with the standard branch count multiplied by $eta$ (approx 112) plus a small overhead, rather than the text's explicit formula. Readers might note this discrepancy, and it would be helpful to specify whether $2^{k+r}$ refers to branches per state or total branches, or if the formula applies to the partition decoding step specifically.

**Status**: [Pending]

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## Detailed Comments (20)

### 1. Clarification Needed on Decoding Complexity Formula

**Status**: [Pending]

**Quote**:
> s complexity is $\beta 2^{k + r}$, where $\beta = 2 - 2^{-k}$, and $2^{k + r}$ is the number of branches per stage of th

**Feedback**:
Readers might note that for a trellis with $2^e$ states and rate $k/(k+r)$, the total number of branches per stage is typically $2^e \times 2^k$. For the 16-state code, the text formula yields 8 branches, whereas the standard trellis structure implies 64 branches. It would be helpful to specify whether $2^{k+r}$ refers to branches per state or total branches, or if the formula applies to the partition decoding step specifically.

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### 2. Ambiguity in 'purely geometric parameters' claim

**Status**: [Pending]

**Quote**:
> The fundamental coding gain of a coset code, as well as other important parameters such as the error coefficient, the decoding complexity, and the constellation expansion factor, are purely geometric parameters determined by $C$ and $\Lambda / \Lambda'$

**Feedback**:
Readers might note that decoding complexity and error coefficient depend on the decoding algorithm and decision regions, not just geometry. It would be helpful to qualify this statement to distinguish between structural properties and implementation-dependent metrics. Rewrite 'are purely geometric parameters' as 'are determined by the code structure and decoding framework' to improve accuracy.

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### 3. Inconsistent Scope of Lattice Definition

**Status**: [Pending]

**Quote**:
> The coset codes described in this paper are based on partitions of binary lattices

**Feedback**:
Readers might note that the opening sentence restricts the paper's scope to 'binary lattices', yet the subsequent text explicitly describes ternary codes and phase-modulated codes as examples of coset codes. This creates an internal inconsistency within the section. It would be helpful to broaden the initial statement to reflect the general group-based framework discussed in the rest of the section.

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### 4. Ambiguity in Ternary Lattice Definition

**Status**: [Pending]

**Quote**:
> where $S$ and $T$ are ternary block codes or lattices, and $C$ is a block or convolutional code over the ternary field GF(3)

**Feedback**:
Readers might note that the text states $S$ and $T$ can be 'lattices' while simultaneously requiring the code $C$ to be over the ternary field GF(3). General lattices do not naturally support codes over GF(3) unless they are specifically constructed from ternary codes. It would be helpful to specify that the lattices must be ternary lattices or constructed from ternary codes.

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### 5. Terminology precision for rotation operator R

**Status**: [Pending]

**Quote**:
> The most important scaled orthogonal transformation for our purposes is the rotation operator $R$, defined by the $2 \times 2$ matrix

**Feedback**:
Readers might note that the matrix $R = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ has a determinant of $-2$, which indicates it includes a reflection rather than being a pure rotation. A rotation matrix must have a positive determinant. It would be helpful to use the term 'scaled orthogonal transformation' consistently or acknowledge the reflection component to maintain linear algebra rigor.

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### 6. Incorrect Set Equality for Integer Lattice Decomposition

**Status**: [Pending]

**Quote**:
> rtition $2Z / 4Z$, and so forth. That is,
> 
> $$
> Z = \left[ Z / 2Z \right] + \left[ 2Z / 4Z \right] + \left[ 4Z / 8Z \right] + \cdots.
> $$
> 
> For a 

**Feedback**:
Readers might note that the set equality is mathematically inconsistent because the right-hand side generates only non-negative integers, whereas $Z$ includes negative integers. It would be helpful to restrict the equality to non-negative integers or clarify the representation for negative values.

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### 7. Typo in Cartesian Product Comparison

**Status**: [Pending]

**Quote**:
> 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 whe

**Feedback**:
Readers might note that the text compares the lattice $\Lambda_N^M$ to itself, which renders the condition on $\gamma(\Lambda_N)$ and $\gamma(\Lambda_M)$ logically vacuous. To make the statement meaningful, the comparison should be between $\Lambda_N^M$ and $\Lambda_M^N$, which are both $MN$-dimensional lattices.

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### 8. Incorrect Definition of Rotation Operator

**Status**: [Pending]

**Quote**:
> the rotation operator $R$ always doubles norms, in any number of dimensions

**Feedback**:
Readers might note that a standard rotation operator preserves vector norms, whereas the text claims it doubles them. This conflates a pure rotation with a scaling transformation. It would be helpful to specify that $R$ includes scaling to maintain mathematical rigor.

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### 9. Confused Volume and Density in Lemma Proof

**Status**: [Pending]

**Quote**:
> the fundamental volume of $\Lambda'$ must be $|\Lambda / \Lambda'| times as dense in $N$-space as those of $\Lambda'$

**Feedback**:
Readers might note that the proof sentence conflates volume with density and compares $\Lambda'$ to itself. It should state that the volume of the sublattice $\Lambda'$ is the index times the volume of the parent lattice $\Lambda$. It would be helpful to rewrite the sentence to correctly express the volume relationship.

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### 10. Notation Error in Sublattice Order Formula

**Status**: [Pending]

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

**Feedback**:
Readers might note that the expression $|\phi|^2\mu$ is ambiguous and mathematically inconsistent if interpreted as multiplication. Since $|\phi|^2 = 2$, the expression $2\mu$ does not equal $2^\mu$ for $\mu > 2$. It would be helpful to rewrite $|\phi|^2\mu$ as $|\phi|^{2\mu}$ to correctly represent the order of the sublattice.

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### 11. Inconsistent Notation for Lattice Depth Parameter

**Status**: [Pending]

**Quote**:
> and the order of $\Lambda / \phi^{n} G^{N}$ is $2^{Nn - r(\Lambda)}$

**Feedback**:
Readers might note that the text previously defines the $\phi$-depth of a complex lattice as $\mu$. However, in this equation, the variable $n$ is used to denote the depth parameter in the exponent. This inconsistency in notation could cause confusion regarding which parameter represents the depth. It would be helpful to standardize the notation to $\mu$ throughout this section.

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### 12. Ambiguous Subscript in $D_k$ 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**:
Readers might note that for the checkerboard lattice $D_k$ in $k$ real dimensions, the normalized redundancy is $2/k$. The text claims $1/2$, which implies $k=4$. Since the previous paragraph explicitly discusses $D_4$, the subscript $k$ appears to be a typo for 4. It would be helpful to correct the subscript to match the specific example discussed.

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### 13. Coset Representative Count Mismatch for 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**:
Readers might note that the text states that the partition $\Lambda_k / \Lambda_{k+1}$ is a two-way partition, which implies it has order 2. However, the text claims the set of $2^K$ binary linear combinations serves as representatives for this specific partition. A set of size $2^K$ cannot represent a partition of size 2 unless $K=1$. It would be helpful to clarify that the combinations represent $\Lambda / \Lambda'$ instead of $\Lambda_k / \Lambda_{k+1}$.

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### 14. Inconsistent depth variable in code formula

**Status**: [Pending]

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

**Feedback**:
Readers might note that the section defines the lattice depth as $\mu$ in the opening sentence. However, the primal lattice formula uses $\phi^{n} \boldsymbol {G} ^ {N}$. Since $n$ is not defined in this section and the depth is explicitly $\mu$, this creates a notation inconsistency. It would be helpful to rewrite the primal formula to use $\mu$ consistently.

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### 15. Table III Dimension Mismatch for D8

**Status**: [Pending]

**Quote**:
> |  D8 | RD4 | 4 | 4 | 2 | 1/2 | 1/2 | 2 | 4 | 10 | 2.5  |

**Feedback**:
Readers might note that the first column lists the lattice $D_8$, but the third column specifies the dimension $2N$ as 4. The $D_8$ lattice is inherently 8-dimensional, whereas $2N=4$ corresponds to a 4-dimensional lattice like $D_4$. It would be helpful to align the lattice name with the dimension.

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### 16. Partition Order Calculation for Z8/2E8

**Status**: [Pending]

**Quote**:
> |  Z8 | 2E8 | 8 | 2^{12} | 4 | 1 | 0 | 1 | 16 | 7376 | ~2  |

**Feedback**:
Readers might note that the table lists the order $|Z8/2E8|$ as $2^{12}$. Given that the volume of $Z^8$ is 1 and the volume of $2E_8$ is $2^8 \times V(E_8) = 256$, the index should be $256/1 = 2^8$. The value $2^{12}$ is inconsistent with standard volume scaling for these lattices. It would be helpful to rewrite the order value to reflect the correct index.

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### 17. Convolutional Code Algebraic Structure Definition

**Status**: [Pending]

**Quote**:
> a(D)$. (It follows that a convolutional code is a vector space over
> 
> the field of binary formal power series $f(D), f_{i} \in \{0,1\}$; the dimension of this v

**Feedback**:
Readers might note that a convolutional code is formally defined as a submodule over the ring of polynomials $F_2[D]$, not a vector space over the field of formal power series $F_2((D))$. A vector space requires closure under multiplication by any field element, including series like $1/(1+D)$, which corresponds to division and is not guaranteed to yield a valid codeword. It would be helpful to rewrite 'vector space over the field of binary formal power series' as 'module over the ring of binary polynomials' to ensure algebraic precision.

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### 18. Inconsistency in Volume Equation Chain

**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**:
Readers might note that the equation chain asserts that $2^{-k(C)}V(\Lambda') = 2^{k+r}V(\Lambda')$, which implies $2^{-k(C)} = 2^{k+r}$. This is algebraically impossible for positive integers $k$ and $r$. It would be helpful to rewrite the equation chain to remove the intermediate term $2^{k+r}V(\Lambda')$ which contradicts the definition of $V(\Lambda')$.

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### 19. Discrepancy in Coding Gain Definition

**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

**Feedback**:
Readers might note that the expression defines $\gamma(\Lambda)$ using $d_{\min}^{2}(\Lambda^{\prime})$, the minimum distance of the sublattice, whereas the coding gain of a lattice $\Lambda$ should depend on its own minimum distance $d_{\min}^{2}(\Lambda)$. This differs from the earlier derived expression which treats $\gamma(\Lambda)$ as the baseline gain for $\Lambda$. It would be helpful to rewrite the expression to use $d_{\min}^{2}(\Lambda)$.

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### 20. Incorrect lattice partition chain indices

**Status**: [Pending]

**Quote**:
> $Z^2 / RZ^2 / 2Z^2 / 2RZ^2 = G / \phi G / \phi^3 G / \phi^3 G$

**Feedback**:
Readers might note that the partition chain on the right side of the equality contains a repetition of $\phi^3 G$ and skips $\phi^2 G$. In the Gaussian integer lattice $G$, the sublattice $2Z^2$ corresponds to $\phi^2 G$. The correct chain should be $G / \phi G / \phi^2 G / \phi^3 G$. It would be helpful to rewrite the chain to reflect the correct indices.

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