Recognition: no theorem link
Angle Between Two Vectors over Finite Fields and an Application to Projective Unique Decoding
Pith reviewed 2026-05-13 03:46 UTC · model grok-4.3
The pith
A Hamming angular function on nonzero vectors over finite fields satisfies metric axioms up to scaling, descends to a metric on projective space, and yields a projective unique-decoding theorem for linear codes.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The author defines angle_H(u,v) as the minimum Hamming distance between u and any nonzero scalar multiple of v. This function satisfies the three metric axioms up to scalar multiplication and is invariant under nonzero scalar multiplications, hence induces an integer-valued metric on the projective space. For a linear code C with minimum distance d, the condition angle_H(u, C excluding zero) less than d/2 guarantees that the closest direction in C to u is unique up to nonzero scalar multiplication.
What carries the argument
The angle_H function, which takes the smallest Hamming distance from one nonzero vector to any scalar multiple of the other.
If this is right
- Linear codes admit a unique closest direction whenever the angular distance to the code is below half the minimum distance.
- Unique decoding can be restated and performed entirely in the projective space of directions rather than on specific vectors.
- The angular formulation connects directly to the proximity-gap analysis of Reed-Solomon codes by replacing ordinary distance with this projective version.
Where Pith is reading between the lines
- The projective invariance may allow decoding procedures that ignore overall scaling and operate only on normalized representatives.
- Similar angular constructions could be examined in other algebraic settings where directions rather than absolute vectors matter.
- The metric property opens the possibility of importing geometric algorithms from real or complex projective spaces into the finite-field case.
Load-bearing premise
The newly defined angular function must satisfy the three metric axioms up to scalar multiplication and remain unchanged under nonzero scalar multiplication in either argument.
What would settle it
A concrete counterexample would be two distinct directions in a linear code C that both achieve the same minimum angle_H distance to some vector u when that minimum is strictly less than d/2, or three vectors where angle_H violates the triangle inequality after scaling.
read the original abstract
We introduce a Hamming-type angular function $$\mathrm{angle}_H(u,v):= \min_{c \in \mathbb{F}_q^n} d_H(u, cv)$$ on pairs of nonzero vectors in $\mathbb{F}_q^n$ and show that it satisfies all three metric axioms up to scalar multiplication. The function $\mathrm{angle}_H$ is invariant under nonzero scalar multiplication in either argument and therefore descends to a genuine integer-valued metric on the projective space $\mathbb{P}(\mathbb{F}_q^n)$. As a concrete application, we prove an \emph{angular} (or \emph{projective}) version of the unique-decoding theorem for linear codes: if $\mathrm{angle}_H(u, C\setminus\{0\}) < d/2$, where $d$ is the minimum distance of the linear code $C$, then the closest direction in $C$ to $u$ is unique up to nonzero scalar multiplication. We then discuss how this angular viewpoint relates to the proximity-gap programme for Reed--Solomon codes. To the best of our knowledge, this is the first attempt to define an angle notion for vectors over finite fields and interpret it from several perspectives, including geometry, coding theory, and cryptography.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces the angular function angle_H(u,v) := min_{c in F_q^*} d_H(u, c v) on nonzero vectors in F_q^n, proves that it satisfies the three metric axioms up to scalar multiplication, establishes invariance under nonzero scalar multiplication in either argument (hence descends to a metric on the projective space P(F_q^n)), and derives the projective unique-decoding theorem: if angle_H(u, C minus {0}) < d/2 for a linear code C with minimum distance d, then the closest direction in C is unique up to nonzero scalar multiplication. The work also relates the construction to the proximity-gap program for Reed-Solomon codes.
Significance. If the central claims hold, the paper supplies the first explicit angle notion on finite-field vector spaces and shows that the projective unique-decoding statement follows directly from the Hamming triangle inequality applied to optimally scaled codewords. The direct, parameter-free verification of non-negativity, symmetry, and the triangle inequality, together with the invariance argument, constitutes a clear technical contribution that may be useful for projective formulations of decoding and cryptographic problems.
minor comments (2)
- [Abstract] Abstract: the minimization is written as min_{c in F_q^n} d_H(u, cv); the correct range is the nonzero scalars F_q^* (as required for scalar multiplication cv). This is a typographical error that should be corrected.
- [Abstract] The phrase 'satisfies all three metric axioms up to scalar multiplication' is used without an explicit statement of what the 'up to' qualifier means for each axiom; a short clarifying sentence or footnote would improve readability.
Simulated Author's Rebuttal
We thank the referee for the careful summary of our manuscript, the positive assessment of its significance, and the recommendation for minor revision. We are pleased that the introduction of the angular metric on finite-field projective space and the resulting projective unique-decoding theorem were viewed as a clear technical contribution.
Circularity Check
No significant circularity
full rationale
The paper introduces angle_H(u,v) by explicit definition as min over scalars of the Hamming distance, then verifies the three metric axioms directly from the corresponding properties of d_H (non-negativity, symmetry via reciprocal scalars, triangle inequality by composing optimal scalings). The projective unique-decoding theorem is obtained verbatim by applying the Hamming triangle inequality to the two optimally scaled closest codewords; no parameter is fitted, no result is renamed, and no load-bearing step relies on self-citation or prior ansatz. The derivation chain is therefore self-contained against the standard Hamming metric and does not reduce to its own inputs.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption angle_H satisfies the three metric axioms up to scalar multiplication
invented entities (1)
-
angle_H function
no independent evidence
Reference graph
Works this paper leans on
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[1]
E. Ben-Sasson, D. Carmon, U. Hab¨ ock, S. Kopparty, S. Saraf,On Proximity Gaps for Reed– Solomon Codes, Cryptology ePrint Archive, Paper 2025/2055, 2025. https://eprint.iacr.org/2025/2055
work page 2025
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[2]
V. Guruswami et al.,Optimal Proximity Gaps for Subspace-Design Codes and (Random) Reed– Solomon Codes, arXiv:2601.10047, 2026. https://arxiv.org/abs/2601.10047
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[3]
S. Chatterjee, P. Harsha, M. Kumar,Deterministic List Decoding of Reed–Solomon Codes, arXiv:2511.05176, 2025. https://arxiv.org/abs/2511.05176
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[4]
B. E. Diamond, A. Gruen,Proximity Gaps in Interleaved Codes, IACR Communications in Cryptology, 1(4), 2025
work page 2025
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[5]
M. P. do Carmo,Riemannian Geometry, Mathematics: Theory & Applications, Birkh¨ auser, Boston, 1992. T ¨UB˙ITAK B ˙ILGEM UEKAE, Gebze, Kocaeli, T ¨urkiye Email address:kamil.otal@gmail.com
work page 1992
discussion (0)
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