# Coset Codes—Part I: Introduction and Geometrical Classification

**Date**: 03/06/2026
**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.

**Outline**

The paper presents a geometrical classification framework for constructive coding techniques, primarily focusing on coset codes for band-limited channels. It utilizes parameters such as fundamental coding gain, redundancy, and normalized decoding complexity to compare lattice and trellis codes. The analysis concludes that trellis codes often offer superior performance-complexity trade-offs compared to lattice codes based on effective coding gain metrics.

**Scope of Classification Claims**

The Abstract asserts that practically all known good constructive coding techniques are characterized as coset codes, yet the detailed classification in Sections V and VI primarily focuses on lattice-type constructions while excluding phase-modulated constellations. It would be helpful to qualify the Abstract to reflect the specific focus on lattice-based constructions or include examples of non-lattice group structures to substantiate the broader claim.

**Validity of Effective Coding Gain Heuristic**

The effective coding gain rankings rely on a rule of thumb stating that every factor of two increase in the error coefficient reduces gain by approximately 0.2 dB, which is acknowledged as approximately valid in Section VII. Readers might note that the range of validity for this approximation is not provided, and it would be helpful to include a sensitivity analysis or rigorous derivation to ensure the performance comparisons in Figure 12 are robust across different error probability curves.

**Complexity Definition and Self-Containment**

There is a discrepancy in the definition of normalized decoding complexity between Section III and Section V, and the detailed trellis-based decoding algorithms are deferred to a companion paper. It would be helpful to unify the normalization factor across the text and include a summary of the algorithmic steps or complexity derivation here to ensure the paper stands as a complete reference for the proposed framework.

**Theoretical Consistency in Code Comparisons**

The paper distinguishes between trellis and lattice codes in Section VII while Section IV-C suggests linear trellis codes are infinite-dimensional lattices, creating conceptual ambiguity regarding the comparison categories. It would be helpful to explicitly distinguish between finite-dimensional lattice codes and infinite-dimensional linear trellis codes and acknowledge that the superiority of trellis codes is contingent on the specific decoding algorithms considered.

**Consistency of Normalization and Parameters**

Readers might note variations in how the normalization factor N is handled when transitioning from lattice definitions to trellis code definitions, particularly in Tables IV through X where dimensions vary. It would be helpful to explicitly clarify how the normalization factor is applied to ensure consistent reporting of dB values and facilitate accurate cross-comparison of codes with different dimensionalities.

**Status**: [Pending]

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## Detailed Comments (19)

### 1. Complexity Definition and Self-Containment

**Status**: [Pending]

**Quote**:
> normalized decoding complexity

**Feedback**:
Readers might note a discrepancy in the definition of normalized decoding complexity between Section III and Section V. It would be helpful to unify the normalization factor across the text to ensure the paper stands as a complete reference for the proposed framework.

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### 2. Theoretical Consistency in Code Comparisons

**Status**: [Pending]

**Quote**:
> . In this sense, linear trellis codes are to finite-dimensional lattices as convo

**Feedback**:
Readers might note that Section VII distinguishes between trellis and lattice codes while Section IV-C suggests linear trellis codes are infinite-dimensional lattices, creating conceptual ambiguity regarding the comparison categories. It would be helpful to explicitly distinguish between finite-dimensional lattice codes and infinite-dimensional linear trellis codes and acknowledge that the superiority of trellis codes is contingent on the specific decoding algorithms considered.

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### 3. Inconsistency Between Table IX and Proof Note

**Status**: [Pending]

**Quote**:
>  5.27 | 60 | 300 | 4.49 |
> 
> Note added in proof: J. Chow (private communication) has obtained values of $\tilde{N}_0 = 88$ and $\gamma_{

**Feedback**:
Readers might note a numerical inconsistency between Table IX and the Note added in proof. Table IX lists d^2_min=8 for the 64-state D_4/RD_4 code, while the Note states d^2_min=6 for the same configuration. It would be helpful to update Table IX to match the Note values to ensure consistency across the section.

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### 4. Viterbi Complexity Formula Omits State Dependency

**Status**: [Pending]

**Quote**:
> inary comparisons, so that its complexity is $\beta 2^{k + r}$, where $\beta = 2 - 2^{-k}$, and $

**Feedback**:
Readers might note that the complexity formula beta 2^{k + r} omits the dependency on the number of states 2^s. Table IV shows normalized decoding complexity N_D scaling linearly with 2^s. It would be helpful to revise the description to include the state dependency, e.g., 'complexity is proportional to 2^s beta 2^{k}'.

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### 5. Inconsistency in Effective Coding Gain Calculation

**Status**: [Pending]

**Quote**:
>  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 

**Feedback**:
Readers might note an inconsistency when applying this rule to Table V. For the 256-state one-dimensional Ungerboeck code, gamma = 6.02 dB and N_0 = 4. If the reference N_0 is 4, the penalty should be 0 dB, yielding gamma_eff = 6.02 dB. However, Table V lists gamma_eff = 5.47 dB. It would be helpful to clarify the reference N_0 used for this specific entry or correct the value.

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### 6. Inconsistent Redundancy Value for D8/RE8 Partition

**Status**: [Pending]

**Quote**:
> 75 |
> | $D_8$ | $RE_8$ | 8 | 32 | 3 | 1 | 3/4 | 2 | 8 | 88 | 2.75 |
> | $D_8^\perp

**Feedback**:
Readers might note a discrepancy in the normalized redundancy rho for the partition D_8/RE_8 in Table III. The text defines rho(partition) = rho(lattice), and Table II lists rho(D_8) = 1/4. Other entries for D_8 in Table III also use rho = 1/4. However, this row lists rho = 3/4. It would be helpful to align this value with the definition and other entries to ensure consistency.

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### 7. Imprecise Terminology for Operator R

**Status**: [Pending]

**Quote**:
> 2$ and its sublattice $RZ^2$ (black dots).
> 
> The most important scaled orthogonal transformation for our purposes is the rotation operator $R$,

**Feedback**:
Readers might note that the term 'rotation operator' applies to the matrix R defined in the text. Calculating the determinant of the 2x2 matrix R yields det(R) = -2. A scaled orthogonal transformation representing a pure rotation should have a positive determinant for its orthogonal component. It would be helpful to use the more general term 'scaled orthogonal transformation' to maintain mathematical precision.

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### 8. Order of Sublattice mZ^N for Negative m

**Status**: [Pending]

**Quote**:
> rs).
> 
> More generally, for any $m \in \mathbb{Z}$, the lattice $mZ^{N}$ of all $N$-tuples of integer multiples of $m$ is a sublattice of $Z^{N}$ of order $m^{N}$, and $[Z / mZ

**Feedback**:
Readers might note that the order (index) of a sublattice must be a positive integer. If m is negative and N is odd, m^N is negative, which cannot represent an order. It would be helpful to restrict m to positive integers or use the absolute value to ensure the order is always positive.

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### 9. Typo in Cartesian Product Density Comparison

**Status**: [Pending]

**Quote**:
> T I
> 1129
> 
> directly) $\Lambda_N^M$ has a greater or lesser density of points per unit volume than does $\Lambda_N^M$ according to whet

**Feedback**:
Readers might note that the text compares Lambda_N^M to itself in the density comparison within MN-space. To compare lattices of dimension N and M fairly, one must compare Lambda_N^M to Lambda_M^N. It would be helpful to rewrite 'than does Lambda_N^M' as 'than does Lambda_M^N' to correct the comparison.

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### 10. Inaccurate Definition of Rotation Operator R

**Status**: [Pending]

**Quote**:
>  $Z^2$ with $d_{\min}^2 = 2$ (the rotation operator $R$ always doubles norms, in any number of dimensions). T

**Feedback**:
Readers might note that an integer matrix that doubles norms (R^T R = 2I) does not exist for all dimensions (e.g., N=3), so the claim 'in any number of dimensions' is inaccurate for integer lattices. It would be helpful to clarify that such integer-preserving operators are dimension-dependent.

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### 11. Inconsistent Variable n vs mu for Complex Depth

**Status**: [Pending]

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

**Feedback**:
Readers might note that the text defines the complex depth of a binary lattice as mu (e.g., 'phi^{mu}G^{N} as a sublattice'). However, in the formula for the order of the partition Lambda / phi^{n} G^{N}, the variable n is used instead of mu. It would be helpful to rewrite 'phi^{n} G^{N}' and 'Nn' as 'phi^{mu} G^{N}' and 'Nmu' to maintain consistency.

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### 12. Typo in Partition Index Notation

**Status**: [Pending]

**Quote**:
> k + r}$, $|\Lambda' / \phi^{\mu} G^{N}| = 2[k(\Lambda')]$, and 

**Feedback**:
Readers might note that the notation 2[k(Lambda')] suggests multiplication, but the context of binary lattice partitions requires the index to be a power of 2. Later in the proof, the text correctly refers to 'one of the 2^{k(Lambda')} cosets'. It would be helpful to rewrite '2[k(Lambda')]' as '2^{k(Lambda')}' to maintain consistency.

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### 13. Algebraic Structure Terminology

**Status**: [Pending]

**Quote**:
> volutional code is a vector space over
> 
> the field of binary formal power series $f(D), f_{i} 

**Feedback**:
Readers might note that the set of formal power series F_2[[D]] typically forms a ring, not a field, because elements like D do not have multiplicative inverses within the set. The text earlier defines the series allowing t to be any integer, which corresponds to formal Laurent series. It would be helpful to distinguish between the ring of power series and the field of Laurent series.

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### 14. Inconsistent Use of mu and rho for Redundancy

**Status**: [Pending]

**Quote**:
> bb{C})$ to compute the fundamental coding gain $\gamma(\mathbb{C}) = 2^{-\mu(\mathbb{C})}d_{\min}^2(\mathbb{C})$, where the normalized redundancy is simply the sum of the normalized redundancies of the code $

**Feedback**:
Readers might note that earlier in the section, rho is defined as the normalized redundancy (rho(C) = 2r(C)/N), and Equation block 2 uses rho(C) for normalized redundancy. However, the final paragraph defines normalized redundancy as mu(C), which was previously defined as the depth of the lattice partition. It would be helpful to use rho(C) consistently for normalized redundancy.

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### 15. Trellis State Description Contradiction

**Status**: [Pending]

**Quote**:
> er $C$ disappears. The trellis associated with $\Lambda^{\infty}$ has one state for each time $t$, and the b

**Feedback**:
Readers might note that the text describes the trellis for the trivial lattice code Lambda^{infty} as having 'one state for each time t'. This implies a time-varying trellis with an infinite number of states, which contradicts the standard definition of a time-invariant trellis code. It would be helpful to correct the phrase 'one state for each time t' to 'a single state' to maintain consistency.

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### 16. Inconsistent Scope Claim in Opening Sentence

**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 claims that the coset codes described in the paper are based on partitions of binary lattices. However, the subsequent text in this section explicitly describes ternary codes, general lattices, and phase-modulated codes (PSK), which are not binary lattices. It would be helpful to broaden the opening statement to reflect the general nature of the coset code framework presented.

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### 17. Inconsistent Notation for Integer Lattice

**Status**: [Pending]

**Quote**:
> mg-4.jpeg)
> Fig. 4. Lattice $Z^2$ and its sublattice $RZ^2$ (black dots).
> 
> The most im

**Feedback**:
Readers might note that the notation for the integer lattice varies between the main text and the figure caption. The text consistently uses the blackboard bold symbol Z (e.g., Z^2), while the caption uses the italic Z (e.g., Z^2). It would be helpful to align the caption with the main text notation to ensure clarity.

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### 18. Ambiguous Lattice Notation in Distance Comparison

**Status**: [Pending]

**Quote**:
> Z^2 / RZ^2 / 2Z^2 / 2RZ^2 / 4Z^2

**Feedback**:
Readers might note that the notation RZ^2 is used without definition in this section to denote a specific lattice partition level. While standard in some contexts, the symbol R is not defined here, making the distance comparison difficult to verify independently. It would be helpful to define R explicitly or use standard notation like D_2 to ensure the mathematical comparison is transparent.

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### 19. Inconsistent Definition of Fundamental Volume

**Status**: [Pending]

**Quote**:
> nd the fundamental volume $V(\mathbb{C})$ per $N$ dimensions, which is equal to $2^{r(\mathbb{C})}$ wh

**Feedback**:
Readers might note that the text defines the fundamental volume V(C) as 2^{r(C)}. However, in the Ungerboeck code example, the sublattice 2Z^2 has a determinant (fundamental volume) of 4, while 2^{r(C)} evaluates to 2. This suggests V(C) refers to the constellation expansion factor rather than the lattice fundamental volume. It would be helpful to clarify that V(C) represents the normalized volume expansion factor relative to the uncoded system.

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