Comments on the recent improvements of the MRRW bounds

• Two concurrent works, [2] recently made a small but conceptually important improvement to MRRW bounds that had resisted improvement for nearly half a century.[18] also improves the Kabatiansky–Levenshtein bound and the sphere-packing exponent, while [2] gives a new upper bound for low-density parity-check codes.Source: §1 · p. 1
• The note focuses on MRRW-1 because its concepts suffice to show the new ideas, although MRRW-2 is numerically better for 0 < δ < 0.273.The Johnson case is more involved, and also uses similar ideas to improve the Kabatiansky–Levenshtein bound for spherical codes.Source: §1 · pp. 1–2
• The note claims no novelty beyond its presentation and contextual remarks, building the proof in from the one-dimensional case of [6] and connecting it to the decoding-based proof in [2].The connection between the Delsarte-type argument and the classical-quantum channel proof is discussed in Section 7.Source: §1 · p. 2
• The same slogan—“move from packing vectors to packing subspaces, and move the subspace”—applies to both proofs [2], and their subspaces are closely related.Source: §1 · p. 2
• The new construction replaces stabilizer-fixed Fourier-level lines with equivariantly moving copies of degree-k Boolean harmonics, whose exponentially large dimension accounts for the asymptotic improvement.At the functional level, it replaces the squared Christoffel–Darboux kernel of the attached lines with the trace kernel of projections onto the moving subspaces.Source: §1 · p. 2

2. THE RESULT

• RMQC(δ) = κH(δ) for 0 < δ < 1/2, so the bounds of and [2] are identical.Source: §2 · pp. 2–3
• The MQC bound traverses the boundary ΓH(a, b) = 1 − 2δ of the OAI optimization with the same objective, whose endpoints give MRRW-1 and the Bassalygo–Elias bound, [14, p.93].Source: §2 · p. 3
• For fixed δ in (0, 1/2), the infimum defining κH(δ) is attained on ΓH(a, b) = 1 − 2δ; κH(δ) ≤ M1(δ), with equality only on b = 0.Source: §2 · p. 3
• The equality of the two new bounds is not a coincidence; the paper promises its explanation in Section 7 after introducing a harmonic-analytic framework.Source: §2 · p. 3
• For δ ≤ 0.0744, the minimum in (2) occurs at r = 0, and RMQC(δ) = RBE(δ).Source: Table 1 · p. 3

3. PRELIMINARIES

• The cube is written as the group {±1}^n, whose Walsh characters form a basis of functions on the cube.With the normalized inner product, the characters are orthonormal; the isometry group is B_n = {±1}^n ⋊ S_n.Source: §3 · pp. 3–4
• Each translated vector v_i,x spans the line in Fourier level V_i fixed by the stabilizer of x.The Hamming distance is expressed as an inner product in the multiplicative notation.Source: §3 · p. 4
• The coordinate-wise decomposition separates t(x,y) into inner products, enabling a Gram factorization and positive-definite kernels.The factor W is included to realize products of t(x,y) with inner products of associated vectors.Source: §3 · p. 5
• The coordinate maps from V_(i+1) and V_(i−1) into W ⊗ V_i are isometries with orthogonal ranges.Their adjoints describe the components in the neighboring levels.Source: §3 · p. 5
• The embedding ∇ maps V_i into W ⊗ (V_(i−1) ⊕ V_(i+1)) and splits into lowering and raising components.For unit f in V_i, their squared norms are determined by the counts i and n−i of coordinates that lower and raise the level, respectively.Source: §3 · p. 5

4. THE CLASSICAL BOUND: ONE LINE PER CODEWORD

• The proof rewrites MRRW-1 [6] on the full space rather than in radial form, a step that becomes essential for the new bound.The presentation is LP-free, with linear programming and orthogonal polynomials left in the background.Source: §4 · p. 6

4.2. The field of lines.

• Each codeword receives a unit vector defining an equivariant field of lines; the resulting kernel is nonnegative and depends only on Hamming distance.At harmonic degree zero, the field’s matrix J(0) represents the classical three-term recurrence; for higher degrees, the two halves no longer reassemble and the Christoffel–Darboux structure is lost.Source: §4.2 · p. 7

4.1. Transition coefficients.

• The transition coefficients between adjacent weight levels are the Ehrenfest urn chain’s transition probabilities, with dimension balance giving detailed balance under the binomial stationary measure.Source: §4.2 · p. 7

4.3. The certificate.

• Certificate positivity depends only on the transition coefficients and the Perron relation, not on harmonic degree, enabling reuse in the generalization.The original certificate of [6] differs slightly and gives a non-exponentially sharper bound.Source: §4.3 · p. 8

4.4. The bound.

• Theorem 4.6 derives the full-space MRRW-1 bound for codes of minimum distance δn when Λ > 1−2δ.The proof averages the certificate kernel and uses its nonnegative Fourier coefficients and a rank bound; avoiding Cauchy–Schwarz recovers a slightly sharper, non-exponential bound from [6].Source: §4.4 · pp. 8–9

5. THE IMPROVED BOUND : ONE SUBSPACE PER CODEWORD

• For k = 0, H0 is the constants and φi,x is the zonal vector vi,x; the higher-degree Boolean harmonics satisfy the stated vanishing-sum condition [19, Sec.II].The classical case of Section 4 corresponds to k = 0.Source: §5 · p. 9

5.1. The tridiagonal matrix, again.

• The tridiagonal-matrix subsection defines Q_i = (i − k + 1)(n − i − k) for k ≤ i < n/2 and gives transition coefficients between adjacent levels.For k > 0, the transition coefficients are sub-stochastic, so the associated Markov chain is not conservative.Source: §5.1 · p. 10, §5.2 · p. 10

5.2. The moving subspace and the bound.

• Replacing a line with a moving subspace preserves a two-point scalar certificate: K(x, y) = tr(PxPy) depends only on Hamming distance.The subspace Ex = im Ψx has dimension dE and moves equivariantly with the codeword: ρ(g)Ex = Egx.Source: §5.2 · p. 10
• When Λ > s, Theorem 5.1 gives a projection bound for every code with minimum distance at least δn.Its proof reuses Theorem 4.6’s argument, changing the diagonal projection value from 1 to dE.Source: §5.2 · p. 11
• The projection bound is interpreted as packing dE-dimensional subspaces in an ambient space of dimension D, with inequalities bounding near-orthogonality and accounting for overlap.If the subspaces were pairwise orthogonal, their number would be at most D/dE.Source: §5.2 · p. 11

6. ASYMPTOTICS

• For k/n→b and L/n→a with 0≤b<a≤1/2, Lemma 6.1 gives A=a(1−a), B=b(1−b), and A−B=(a−b)(1−a−b).Source: §6 · p. 11
• The proof bounds λmax above by the maximum row sum using Perron-Frobenius and obtains a matching asymptotic lower bound via Rayleigh-Ritz.For the lower bound, it uses a unit vector supported on the last m=o(n) coordinates, each equal to 1/√m.Source: §6 · p. 11
• 1/n log2 D_d^E → h2(a) − h2(b), yielding Eq. (1) when Γ_H(a,b) > 1 − 2δ and k/n→b, L/n→a.Under this condition, Lemma 6.1 ensures the hypothesis Λ > s of Theorem 5.1 holds for all sufficiently large n.Source: §6 · p. 11

7. THE PROBABILISTIC PICTURE: CLASSICAL-QUANTUM CHANNELS

• The pure-state channel [2] attaches a line to each codeword, with pairwise overlaps determined by Hamming distance.The line family is equivariant under the full isometry group of the Hamming cube.Source: §7 · p. 12
• The average output spectrum is binomial across Fourier levels, and its typical subspace has dimension 2^((h2(q)+o(1))n), serving as the fixed ambient space.Its dimension is identified with 2^((h2(a)+o(1))n) after setting a = q.Source: §7 · pp. 12–13
• The mixed-qubit channel leaves the ambient space unchanged, while each codeword’s conditional typical subspace moves with it and has dimension 2^((h2(b)+o(1))n).This moving subspace is the probabilistic counterpart of the equivariant field x ↦ E_x ⊂ V.Source: §7 · pp. 13–14
• The criterion in [2] chooses η so the PGM bit error rate equals δ; the optimal channel is strictly mixed, with η > 0.At η = 0 the state is pure and the attached object is a line; for η > 0 it is a higher-dimensional subspace.Source: §7 · pp. 13–14
• Both proofs pack subspaces of asymptotically the same dimension, using exact positive-definite trace counting in and typicality with the classical-quantum strong converse in [2].The probabilistic packing is interpreted through a packing visualization.Source: §7 · p. 14
• Both and [2] strictly improve MRRW-2; uses constant-weight Johnson-scheme spaces, while [2] uses a masking channel that reveals parity.The mask pattern plays the role of the layer, and the revealed parity carries the Bassalygo–Elias reduction.Source: §7 · p. 14

8. COMPLEMENTS AND FURTHER REMARKS

• Moving a subspace with each codeword preserves a scalar two-point certificate, unlike the matrix-valued three-point approach for fixed base points; obtains it through coordinate-wise Gram factorization.The factorization yields a strictly smaller eigenvalue, while the usual multiplication-by-distance attempt and the positivity mechanism of do not extend to k > 0.Source: §8 · pp. 14–15
• The classical radial construction uses a Perron eigenvector to form the Christoffel-Darboux kernel and a product of positive-definite kernels, yielding an exponentially unchanged bound; [6] gives more details.The zonal vector is a Krawtchouk polynomial of distance, and the tridiagonal matrix is a radial version of the Hamming-ball adjacency matrix.Source: §8 · p. 15
• For k > 0, transition coefficients are not mass-preserving, so the retained family has no three-term recurrence, and no individual certificate summand is Delsarte-feasible.The full certificate is not a single-polynomial square of the family optimized in; an explicit form and optimization over sums of squares remain open questions.Source: §8 · p. 15
• The spectral approach extends to Johnson spaces and the real sphere, where the OpenAI extension strictly improves the classical asymptotic bound; also improves MRRW-2 for constant-weight and general binary codes.Possible improvements in projective spaces, q-ary Hamming space, Grassmann manifolds, and ordered Hamming space are expected to be more technically involved and potentially less consequential.Source: §8 · p. 15

8.1. Where a further improvement would come from.

• A possible further improvement could come from reducing the spectral-radius gap between multiplication by distance and J(k), whose matrices differ for k > 0.The remarks are speculative rather than the outcome of systematic investigation, and no proof establishes that these directions yield further improvements.Source: §8.1 · pp. 15–16

8.1.3. For ℓ≥2 let

• At ℓ = 2, RMQC is best possible, while at ℓ = 4 the masked PSC recovers the second MRRW bound and masked MQC improves it; the limit as ℓ grows remains unknown.Further improvement could use a better decoder or different output states, since the PGM error rate is only a sufficient criterion and bitwise optimality holds only for linear codes.Source: §8.1.3 · p. 16
• A better output-symmetric channel would not immediately yield a Delsarte certificate: its positive kernel is nonnegative everywhere, whereas the certificate must be nonpositive at distances at least δn.The channel argument lacks the sign-changing and spectral-radius ingredients needed for the construction; it is asymptotic through typical subspaces, unlike the exact finite-n Delsarte proof.Source: §8.1.3 · pp. 16–17