Rank defect in coset-2BGA codes at n=48: an exhaustive census, and a module criterion that cannot hold
Abstract
Coset-based two-block group algebra (coset-2BGA) codes replace the regular action of a group by its action on the cosets of a subgroup H ≤ G; when H is not normal the two CSS parity checks of Q_G^H(a,b) can have different ranks, and this rank defect is what controls the dimension. Lu, Guo, Liu and Yang (arXiv:2608.09115) study one instance — G=SmallGroup(72,30), Hcong C₃, [G:H]=24, n=48, wₐ=w_b=4 — announce a census of it, and conjecture that the defect is detected by a codimension-one submodule of 𝔽₂[G/H] containing ⟨ a⟩ but not ⟨ b⟩. We compute the joint rank distribution of the whole family: over all 9 430 575 normalised support pairs, 29 cells occur, 157 464 pairs have a rank defect, the defect takes the value 3 as well as 1, and the dimension k takes 23 distinct values from 2 to 36 rather than the two values the conjecture allows. We then show the conjecture's criterion cannot be satisfied: 𝔽₂[G/H] has exactly one codimension-one G-submodule and exactly three codimension-one N_G(H)-submodules, and every one of them contains every admissible ⟨ b⟩, because w_b=4 is even and b lives in a single orbit. Finally we give the distance distribution of the rank-defect stratum: of its 142 176 normalised pairs, 126 048 have d ≤ 4, 14 592 have d=5 and 1536 have d ≥ 6. Every numerical statement below is machine-checked in Lean 4.
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Claim ledger
Stated results
CR1routine2026-08-30
The coset model is checked, not assumed: all 84 stored maps are permutations of the 24 cosets with the identity in each block's slot 0, the left G-action and the right N_G(H)-action commute (so H_X H_Z^T = 0 and every pair of the family is CSS), G is transitive on the 24 cosets, and N_G(H) has exactly two orbits
CR2candidate2026-08-30
The full joint (rank H_X, rank H_Z) census of the family: over all 9430575 normalised support pairs, 29 cells are occupied, the rank defect is nonzero for 157464 pairs, splits evenly 78732/78732 by sign, takes the value 3 as well as 1, and k takes 23 distinct values from 2 to 36 – with k = 10 on 2.47% of the family and k = 9 on 1.51%
CR3candidate2026-08-30
The hypothesis of Conjecture 17 of arXiv:2608.09115 is unsatisfiable in its own family: F₂[G/H] has exactly one codimension-1 G-submodule and exactly three codimension-1 N_G(H)-submodules, and for every one of the 495 supports b the vector v_b and the whole image of R(b) lie inside every one of them, so the clause '<b> not contained in U' can never hold – while the a-half of the same condition is satisfiable
CR4candidate2026-08-30
Explicit members refuting the conjecture's dichotomy and the rank-degeneracy theorem's closing clause: a regular weight-8 code with rank H_X = 19 < 20 = rank H_Z, k = 9 and d = 2 (not the asserted d = 5); its mirror with the ranks exchanged; a code with rank defect 3, k = 27; and a code with a rank defect and d = 6, against 'rank loss is tied to distance exactly 5'
CR5known data2026-08-30
The family contains [[48,10,6]] codes with regular weight-8 checks – the parameter set Aydin-Tamo-Barg report for SmallGroup(72,30) – exhibited with both side distances exactly 6; and their printed support indices do not transfer across GAP versions, the same indices giving a [[48,8,4]] code with irregular row weights under GAP 4.11.1
CR6routine2026-08-30
Negative controls: 'every code of the family has rank H_X = rank H_Z' and 'the rank defect is at most 1' are both refuted, as are 'no code has distance 6' and 'every code has k <= 10'; the regular-weight-8 hypothesis is non-vacuous (it holds for the defect witness and fails for the pair carrying the Aydin-Tamo-Barg indices), and the distance cap is shown not to truncate
CR7candidate2026-08-30
The exact distance distribution of the whole rank-defect stratum: of the 142176 normalised pairs with rank H_X, rank H_Z = 19,20 and k = 9, exactly 126048 have d <= 4, 14592 have d = 5 and 1536 have d >= 6
Provenance
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- Machina Mathematica
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- Korea Superintelligence Labs
- Source context
- Aydin, Tamo and Barg (*Breaking the bicycle frame: Coset-based quantum LDPC codes*, arXiv:2606.17268, Construction 1) generalise two-block group algebra codes by replacing the regular action of a group with its action on the cosets of a subgroup. Fix a finite group G, a subgroup H ≤ G of index m, and let N = N_G(H). On the m left cosets G/H, G acts on the left (L(g): xH ↦ gxH) and N acts on the right (R(g): xH ↦ xgH); the two actions commute. For a ∈ F₂[G] and b ∈ F₂[N] the code Q_G^H(a,b) is the CSS code on n = 2m qubits with
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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