Optimally Perturbed Identity Matrices of Rank 2
Pith reviewed 2026-05-24 22:32 UTC · model grok-4.3
The pith
For real rank-2 matrices the Bukh-Cox relaxation of the Gram-matrix condition is tight.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
When the matrices are restricted to real entries and rank exactly 2, the smallest ε for which a diagonal-1 matrix with off-diagonal absolute values at most ε exists is identical to the smallest ε permitted by the Bukh-Cox relaxation that drops the Gram-matrix condition.
What carries the argument
The Bukh-Cox relaxation obtained by removing the Gram-matrix (positive-semidefinite) constraint from the search for perturbed identity matrices.
If this is right
- The minimal achievable ε for real rank-2 perturbed identities equals the relaxed minimum for every dimension n.
- Exact optimal perturbations can be constructed by solving the relaxed problem and verifying rank 2.
- The same relaxation supplies sharp bounds on the size of real antipodal codes in two dimensions.
Where Pith is reading between the lines
- The exact match at rank 2 may indicate that the relaxation becomes loose only when rank or field changes.
- One could test whether the same matrices also achieve the bound when the Gram condition is re-imposed after rounding.
- The rank-2 real case supplies a concrete test bed for comparing the relaxed bound against exhaustive search over small n.
Load-bearing premise
The Bukh-Cox relaxation remains a valid bounding technique whose tightness can be exactly characterized when the matrices are restricted to real entries and rank 2.
What would settle it
Exhibit a real rank-2 matrix with diagonal 1 whose largest off-diagonal absolute entry is strictly smaller than the value given by the relaxed problem; or prove that no such matrix exists for a concrete n.
read the original abstract
The problem of optimal antipodal codes can be framed as finding low rank Gram matrices $G$ with $G_{ii} = 1$ and $|G_{ij}| \leq \epsilon$ for $1 \leq i \neq j \leq n$. In 2018, Bukh and Cox introduced a new bounding technique by removing the condition that $G$ be a gram matrix. In this work, we investigate how tight this relaxation is, and find exact results for real valued matrices of rank $2$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript frames optimal antipodal codes as the search for low-rank Gram matrices G with unit diagonal and off-diagonal entries bounded in absolute value by ε. It studies the Bukh-Cox relaxation obtained by dropping the Gram-matrix (positive-semidefinite) constraint and derives exact characterizations of the tightness of this relaxation when the matrices are restricted to real entries and rank exactly 2.
Significance. The exact results supply a concrete, low-dimensional benchmark for the gap between the original constrained problem and the relaxation. This benchmark is useful for assessing the relaxation's utility in coding-theory bounds and may serve as a test case for extensions to higher rank or complex entries. The rank-2 restriction permits an exhaustive solution of the resulting optimization problem.
minor comments (3)
- [Abstract] Abstract: the phrase 'exact results' is used without indicating which quantities (e.g., the minimal achievable ε or the optimal matrix entries) are determined exactly; a single sentence clarifying the output of the analysis would improve readability.
- [Introduction] The connection between the relaxed problem and the original Gram-matrix formulation could be stated as an explicit pair of optimization problems (one constrained, one unconstrained) early in the introduction to make the tightness statement precise.
- Notation: the symbol ε is introduced for the off-diagonal bound but is not consistently distinguished from the optimal value ε* achieved by the relaxation; a brief remark on this distinction would prevent confusion.
Simulated Author's Rebuttal
We thank the referee for the supportive summary, the assessment of significance, and the recommendation of minor revision. The report lists no specific major comments under the MAJOR COMMENTS section.
Circularity Check
No significant circularity
full rationale
The paper examines the tightness of the Bukh-Cox relaxation (introduced by external authors in 2018) when restricted to real matrices of rank exactly 2. The abstract and problem framing present this as a direct comparison between the original Gram-matrix constrained problem and the relaxed version, with exact results obtained via low-dimensional analysis. No self-citations, fitted parameters renamed as predictions, self-definitional steps, or ansatzes imported from the authors' prior work are indicated. The central claim remains independent of the paper's own inputs and is externally benchmarked against the Bukh-Cox technique.
discussion (0)
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