A graph G is k-vertex-minor universal if every graph on every k of its vertices is a vertex-minor of G. Let ν(k) be the least order of such a graph; the first three values are ν(2)=3, ν(3)=6, ν(4)=10, and at k=5 the best published bound is ν(5) ≤ 17, from t…
A tile of a box C=ℤ_(d₁) × … × ℤ_(d_N) is a product R₁ × … × R_N of nonempty, not necessarily consecutive, subsets Rᵢ ⊆ ℤ_(dᵢ). Han, Zhang, Shi and Zhang call a partition C=bigsqcupⱼ₌₁ˢtⱼ into tiles with s ≥ 3 an O_N-tile decomposition when no union bigsqcu…
Call a set S ⊆ [q]ⁿ of words unextendible if every word of [q]ⁿ disagrees with some member of S in every coordinate. The minimum size of such a set carries two names in two literatures that do not cite each other: it is the minimum skirting set f(n,q) of Ad…
The stabilizer rank χ(psi) of an n-qubit state psi is the least number of stabilizer states whose complex span contains psi. For the two Clifford-inequivalent single-qubit magic-state orbits — the octahedron-edge state |H⟩=cos(π)/(8)|0⟩+sin(π)/(8)|1⟩ and th…
In arXiv:2602.20158, Liang and Chen build qudit CSS low-density parity-check codes from a pair of weight-three Laurent polynomials over a twisted two-dimensional torus, and tabulate sixty-five of them at p ∈ {3,5,7,11}. The distance column of those tables i…
A quantum Latin square of order n is an n × n array of unit vectors of ℂⁿ every row and every column of which is an orthonormal basis; its cardinality is the number of entries once vectors differing by a global phase are identified. Zhang, Lv and Cao determ…
The MacWilliams extension theorem fails over the label alphabet A=𝔽₂² of qubit Pauli operators, and Mahmoud has recently located the smallest scales at which it fails for stabilizer codes. Writing k for the 𝔽₂-dimension of the stabilizer, he determines, f…
The kite matrix Aₙ is the n × n zero–one matrix whose (i,j) entry is 1 exactly when i+j ≤ n+1. Gross and Goedicke, constructing doubly perfect functions on the discrete phase space ℤ₂²ⁿ — and through them two-unitary matrices and absolutely maximally entang…
Let S₁,…,Sₙ be Pauli strings on m qubits and let G be their frustration graph, whose edges record the anticommuting pairs. Xu, Wang, Ye, Koßmann, Schwonnek and Winter attach to G and a weight w ∈ ℝⁿ₊ the weighted beta number β(G,w)=sup_ρΣᵢwᵢ⟨ Sᵢ⟩_ρ², call G…
A self-dual additive code over 𝔽₄ of length n is an n-qubit stabilizer state, and every such code is equivalent to the graph code C_G={(x,Γ_G x):x ∈ 𝔽₂ⁿ} of a graph G on n vertices; two graph codes are monomially equivalent — equivalently, the two states…
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…
A quantum code is constant excitation (CE) if its whole code space lies in a single eigenspace of the total excitation operator N=Σⱼ diag(0,…,p-1)ⱼ; such a code acquires only a global phase under the collective coherent drift e^(-iθ N). For qubits N is the…
A quantum code is constant excitation (CE) if every computational basis ket occurring in its code space has the same Hamming weight; such a code acquires only a global phase under collective coherent Z-rotations. Lai, Liou and Ouyang generalise the CSS cons…
A quantum code is constant excitation (CE) if every computational basis ket occurring in its code space has the same Hamming weight; such a code is immune to collective coherent Z-rotations. We determine the exact minimal length in three cases. First, no [[…
For a finite abelian group G and A,B ∈ 𝔽₂[G], the bicycle code of the pair (A,B) is the CSS code on n=2|G| qubits with check matrices H_X=(A B) and H_Z=(B^(T) A^(T)). Surveying more than 170 three-factor tori with weight-3 polynomials, Galimova observed th…
A Hermitian self-dual MDS code [2k,k,k+1]_(q²) produces, through the stabilizer construction, an absolutely maximally entangled state on 2k parties of local dimension q, and such codes are exactly the k × k matrices A over 𝔽_(q²) with Aoverline(A)^(T)=-Iₖ…
