Comment on "Symmetric Pseudo-Random Matrices"

• The matrices are constructed from nonzero codewords of a binary simplex code, called Golomb sequences or m-sequences, with cyclic shifts forming the rows of a binary circulant matrix.The code has parameters [n = 2^m−1, m]₂.Source: §I · p. 1
• The earlier work used simulations and a method-of-moments proof to support semicircle convergence for the matrix ensemble.The simulations used 100 instances at m = 13 and 14.Source: §I · pp. 1–2
• The authors identify technical flaws in the prior proof and show Proposition 2 is false at least for r = 1.Source: §I · p. 2
• The comment expresses the eigenvalues as normalized twisted Kloosterman sums and applies Katz’s result to establish deterministic semicircle convergence.It states this is stronger than the conclusion in [4, Remark 1].Source: §I · p. 2
• The note is organized around the technical flaws and a derivation of the eigenvalue expression.Source: §I · p. 2

• The section examines technical flaws and inappropriate applications of technical lemmas in [4].Source: §II · p. 2

• The mapping from the index tuples to T_r is n-to-one, so the summation sets have cardinalities n^r and n^(r−1), respectively.Adding the same integer modulo n to every i_q preserves t.Source: A. Computation Error · p. 2
• Consequently, the right-hand side of the derived relation is missing a factor of n.Source: A. Computation Error · p. 2

• A non-even cycle can have ν(t) = 0, so the set with ν(t) = 0 is larger than the set of even cycles.For r = 4, the cycle (1, 2, 4, 3) gives t = (1, 2, −1, −2) and ν(t) = 0.Source: B. False Claims · p. 3
• When r is even, the ν(t) = 0 class contributes Gaussian moments as its leading term, rather than the presumed even-cycle classification.The passage gives (r−1)!! n^(r/2) + O(n^(r/2−1)) possibilities.Source: B. False Claims · p. 3
• The claimed uniform bound for ρ_r(ℓ) fails when ℓ < r, and the restricted count depends on the codeword itself, not only its weight.Source: B. False Claims · p. 3
• The asserted divisibility of every codeword weight in H by 4 is false; nonzero weights are only shown to be even and at least 4.The passage says this flaw does not significantly affect the subsequent argument.Source: B. False Claims · p. 3

C. Inappropriate Applications of Technical Lemmas

• Lemma 7 cannot be applied directly to ⟨H′⟩ because it is related to a half-puncturing of S(D, G), and the corresponding G− has degree 2 on a single point.Lemma 6 requires G− to be a single point.Source: C. Inappropriate Applications of… · p. 3
• These issues leave the derivation of III from III′ unsupported by the cited lemmas.Source: C. Inappropriate Applications of… · pp. 3–4
• The relation in Eq. (72) is multiplicative rather than linear over F2 or F2m, so Lemma 6 is inapplicable.The authors' claim that α increases by 1 under this relation therefore does not justify applying the lemma.Source: C. Inappropriate Applications of… · p. 4

D. Order of the Variance in Proposition 2

• O(1) variance does not verify the claimed almost sure convergence in [4, Corollary 2], nor convergence in probability.The right-hand side of Eq. (102) is too small by a factor of n^2.Source: D. Order of the Variance in… · p. 4
• The first-moment variance can be computed directly without a combinatorial argument.The diagonal-entry characterization gives the calculation.Source: D. Order of the Variance in… · p. 4
• 1/(4n) is the variance of β1(An), since it is a Rademacher variable divided by 2√n.β1(An) is any diagonal entry of the circulant matrix, so this directly disproves Proposition 2 for r = 1.Source: D. Order of the Variance in… · p. 4

E. Definition of Standard Semicircle Law

• The standard or standardized semicircle law is defined here to have unit variance.Source: E. Definition of Standard Semicircle Law · p. 4
• Its probability density is specified over [−2, 2].Source: E. Definition of Standard Semicircle Law · p. 4
• Under this convention, the associated random variable is twice that defined in [4, Eq. (21)].Source: E. Definition of Standard Semicircle Law · p. 4

A. Problem Set-up

• The paper uses an algebraic representation of the nonzero codewords of C to formulate the Golomb-sequence construction.Source: A. Problem Set-up · p. 4
• The ensemble A′n uses entries defined by a sign from φa(j−i)+φa(i−j), scaled by 1/√n.The inputs to φa are interpreted modulo n; this normalization is chosen to conform to the standard scale of probability distributions.Source: A. Problem Set-up · p. 4
• The stated goal is convergence of the spectral measure of A′n to the standard semicircle measure as m grows.Source: A. Problem Set-up · p. 4

B. Proof of Theorem 1

• The symmetric circulant structure lets the proof compute the eigenvalues directly instead of using expected spectral moments.The coefficients c_r(a) satisfy c_{n−r}(a) = c_r(a).Source: B. Proof of Theorem 1 · p. 5
• The eigenvalue numerator is the twisted Kloosterman sum Kψa(1, 1; χk), with ψa a non-trivial additive character.The sum is considered over multiplicative characters χk.Source: B. Proof of Theorem 1 · p. 5
• Katz proved that the normalized twisted Kloosterman sums varying over χk are equidistributed for the semicircle measure as q grows [3].The paper applies this result to establish the matrix spectral limit.Source: B. Proof of Theorem 1 · p. 5
• As n grows, the spectra are equidistributed for the semicircle measure on [−2, 2].The proof concludes this for both eigenvalue families.Source: B. Proof of Theorem 1 · p. 5