Spectral Theory of Semisimple Bivariate Bicycle Codes

Orbits, idempotents, and regions

• Frobenius orbits index component idempotents eO, selected-support ideals JS, and active or zero-set regions whose Fourier transforms exchange complementary regions.The notation distinguishes simultaneous, single, and absent support or vanishing of the defining polynomials.Source: Orbits, idempotents, and regions · p. 2

I. INTRODUCTION

• Logical dimension is determined by the common-zero Frobenius-orbit region, while single-zero orbits are homologically trivial and X- and Z-logical spaces are algebraically dual.The defining ring decomposes into orbit components, allowing code parameters to be read from polynomial projections.Source: §I · p. 3

BB codes and distance bounds

• A colon-ideal bound captures mixed-block logical operators and, with alternating stabilizer exclusion, yields a sharper computable lower bound on quantum distance.The bound is expressed through minimum Hamming weights of classical 2D cyclic codes.Source: BB codes and distance bounds · p. 2, §I · p. 3

Later sections

• BB-code automorphisms depend on spectral slopes across active-support orbits, enabling signed block-swapping duality, phase-type CZ gates, and canonical redundant-stabilizer metachecks.The relevant automorphism groups include permutation and local-Clifford–permutation symmetries.Source: Later sections · p. 2, Orbits, idempotents, and regions · p. 2, §I · p. 3

I. INTRODUCTION

• The framework analyzes code lifts through check-ideal commutative algebra and quotient-ring surjections, producing an exact dimension formula from strictly new common roots over any finite field Fq.These lifts connect base codes to covering-graph descendants.Source: §I · p. 3
• For fixed grids, spectral enumeration determines BB-code properties without writing stabilizers, while examples illustrate the method and product constructions support BCH-based distance bounds.The paper emphasizes deterministic understanding rather than numerical-search tables.Source: §I · p. 3

II. MATHEMATICAL PRELIMINARIES & NOTATION

• The setting uses R = Fq[x, y]/⟨xℓ−1, ym−1⟩ with gcd(q, ℓm) = 1, identifies R with Fq[Zℓ × Zm] and Fq^ℓm, and distinguishes product-ring ideals from R-submodules in R × R.Source: §II · pp. 4–5
• The Euclidean pairing is nondegenerate and symmetric with an orthonormal monomial basis, R is symmetric Frobenius, and multiplication satisfies M_c^⊤ = M_ι(c).For every ideal I, ann⟨ann⟨I⟩⟩ = I and dimFq I + dimFq ann⟨I⟩ = ℓm.Source: §II · p. 5

• Cyclic codes of length n correspond bijectively to ideals of Fq[x]/⟨x^n−1⟩, and a code with check-polynomial degree k has parameters [n, k]_q.The generator polynomial divides x^n−1, while the complementary check polynomial defines the code dimension.Source: §III · pp. 6–7
• When gcd(q,n)=1, irreducible factors of x^n−1 are organized by q-cyclotomic cosets, which determine minimal polynomials and defining sets.The defining set completely determines the generator polynomial, and its associated powers of a primitive n-th root form the code’s zero set.Source: §III · pp. 6–7
• The dual of a cyclic code is cyclic, with generator h*(x)/h(0), and its defining set is {−i mod n | i ∉ T_C}.Here h*(x) is the reciprocal of the check polynomial, reflecting inversion of roots under Euclidean duality.Source: §III · pp. 7–8
• A defining set containing δ−1 consecutive exponents guarantees d(C) ≥ δ, so a BCH code with designed distance δ has minimum distance at least δ.BCH constructions choose the smallest union of cyclotomic cosets containing the required consecutive exponents to balance distance and dimension.Source: §III · p. 8
• The Chinese Remainder Theorem decomposes the semisimple ring into field components, whose primitive idempotents act as indicators for cyclotomic-coset frequency components.Idempotents can be computed using the extended Euclidean algorithm and generate the same cyclic ideals as the corresponding polynomial generators.Source: §III · pp. 8–9
• Cyclic-code containment is equivalent to reverse divisibility of generator polynomials and reverse inclusion of defining sets: C1 ⊆ C2 iff g2(x) | g1(x) iff T_C2 ⊆ T_C1.Including or excluding cyclotomic cosets therefore produces a lattice of cyclic codes.Source: §III · p. 10

Hartmann–Tzeng Bound

• The Hartmann–Tzeng bound gives d(C) ≥ δ + s when the defining set contains suitably shifted strings of consecutive roots, reducing to BCH when s = 0.It requires gcd(m1,n)=1 and gcd(m2,n)<δ under the stated indexing conditions, and can be viewed as a multi-string BCH bound.Source: §III · p. 10

Roos Bound

• The Roos bound gives d(C) ≥ δ + s when the zero set contains a product set M N whose consecutive-root set N has at most δ − 2 holes.The condition is |N| − |N| ≤ δ − 2, equivalently is − i0 ≤ δ + s − 2; BCH and Hartmann–Tzeng bounds arise from specific subset choices.Source: §III · p. 11

Shift Bound

• The shift bound lower-bounds codeword weight by the size of an independent set constructed from the codeword’s zero set, yielding d(C) ≥ D.It is described as often stronger than the BCH, Hartmann–Tzeng, and Roos bounds, though less elementary to state.Source: §III · p. 11

Dual BCH Codes

• For primitive binary BCH codes, the Carlitz–Uchiyama theorem uses exponential-sum estimates to restrict nonzero dual-codeword weights and thereby lower-bound the dual minimum distance.The setting is n = 2^m − 1 with designed distance δ = 2t + 1.Source: §III · p. 11

Definitions and motivation

• 2D cyclic codes are ideals in R = Fq[x, y]/⟨x^ℓ−1, y^m−1⟩, where nonprincipal ideals may require multiple generator polynomials.Existing work emphasizes canonical generators and polynomial algorithms such as Gröbner bases, while the semisimple setting permits an idempotent-based theory closer to the 1D case.Source: §IV · p. 11

Reduction to one dimension

• When gcd(ℓ,m)=1, the coordinate group and ring reduce to their 1D counterparts, so every 2D cyclic code is essentially a 1D cyclic code of length ℓm.The homomorphism Φ maps x to z^m and y to z^ℓ, is surjective because Bézout relations place z in its image, and identifies the quotient with Fq[z]/⟨z^ℓm−1⟩.Source: §IV · pp. 11–12

Genuinely 2D case

• A genuinely two-dimensional theory emerges only when gcd(ℓ,m)≠1, because the grid no longer unrolls into a single cyclic coordinate.This is the regime in which the paper’s semisimple 2D cyclic-code theory is applied to bivariate bicycle codes.Source: §IV · p. 11

Semisimplicity assumption

• The ring R is semisimple exactly when gcd(ℓm,q)=1, equivalently when the characteristic of Fq divides neither ℓ nor m.Otherwise R is non-semisimple or repeated-root; the paper assumes the semisimple case and leaves the non-semisimple case for future work.Source: §IV · p. 12

Definitions

• q-Frobenius orbits generalize one-dimensional q-cosets on Z_ℓ × Z_m, with their collection denoted Ω.The orbit decomposition acts on the root-grid index set used for bivariate cyclic codes.Source: A. Orbits · p. 12

Stratified orbit counting

• For strata Λ_d,e classified by coordinate orders d and e, every orbit has length lcm(ord_d(q), ord_e(q)), and each stratum contains φ(d)φ(e)/lcm(ord_d(q), ord_e(q)) orbits.Thus the multiset of orbit sizes is determined by the multiplicative orders of q modulo divisors of ℓ and m.Source: A. Orbits · pp. 13–14

Burnside and Möbius counting

• Burnside’s lemma counts total orbits from fixed points, with Fix(φ^t) = gcd(ℓ, q^t − 1)gcd(m, q^t − 1).The cyclic action generated by φ has order lcm(ord_ℓ(q), ord_m(q)); Möbius inversion further recovers counts by exact orbit length.Source: A. Orbits · pp. 14–15

Semisimple ring structure

• In the semisimple case, q-Frobenius orbits correspond one-to-one with minimal idempotent generators, while orbit lengths determine the degrees of the associated finite-field extensions.Bivariate bicycle-code dimensions and minimum distances depend on how defining polynomials project onto these irreducible components.Source: A. Orbits · p. 15

Example: q = 2, ℓ = m = 3

• For q = 2 and ℓ = m = 3, the index set decomposes into exactly 5 q-Frobenius orbits: one of length 1 and four of length 2.Both stratified counting and Burnside–Möbius calculations yield the same decomposition.Source: A. Orbits · p. 15

Primitive idempotents

• Evaluation identifies R_K with a direct product of ℓm copies of K, whose coordinate projections form the complete set of primitive orthogonal idempotents e_i,j.The construction assumes gcd(q,ℓ)=gcd(q,m)=1 so the defining polynomials are squarefree and split over K.Source: B. Idempotents · pp. 15–16

Frobenius orbits

• q-Frobenius permutes primitive idempotents along the index map (i,j)↦(qi,qj), while summing over each orbit produces an invariant idempotent e_O in R.Orbit idempotents are pairwise orthogonal and sum to 1; individual e_i,j lie in R exactly when ℓ divides i(q−1) and m divides j(q−1).Source: B. Idempotents · pp. 16, 18

Explicit formulas

• Each primitive idempotent has both a Lagrange-interpolation expression L_i(x)M_j(y) and a discrete-Fourier expression characterized by δ_i,i′δ_j,j′ evaluations.The two formulas agree because both have the defining coordinate-evaluation property under the algebra isomorphism.Source: B. Idempotents · pp. 16–17

Wedderburn decomposition

• The map R→∏_{O∈Ω}e_OR is a ring isomorphism, and every component e_OR is the finite field F_q^{|O|} of size q^{|O|}.Thus orbit sizes are exactly the extension degrees in the Wedderburn decomposition.Source: B. Idempotents · pp. 19–20

Module and ideal decompositions

• As an R-module, R decomposes as the direct sum of the ideals e_OR, with dim_Fq(e_OR)=|O|; arbitrary orbit-indexed ideals have dimensions given by summing these orbit sizes.For any subset T⊆Ω, the corresponding ideal is the direct sum of e_OR over O∈T.Source: B. Idempotents · pp. 20–21

C. Lattice Theory

• For S ⊆ Ω, the S-selected idempotent is e_S = ∏_{O∈S} e_O, and its corresponding ideal is J_S = e_SR ◁ R.These definitions introduce the subset-indexed idempotent and ideal construction.Source: C. Lattice Theory · p. 21
• Theorem 54 establishes a bijection between the power set P(Ω) and the ideals of R.The bijection is made precise by representing ideals through the selected-ideal construction.Source: C. Lattice Theory · p. 21
• Every ideal J ◁ R has the form J = J_S for a unique subset S ⊆ Ω.Thus, the selected-ideal representation is both exhaustive and unique.Source: C. Lattice Theory · p. 21

Ideal correspondence

• The ring has exactly 2^|Ω| ideals, naturally bijecting ideals with subsets of q-Frobenius orbits through their spectral idempotents.For an ideal J, its zero set is Frobenius-closed, and J=e_S R for the uniquely determined set S of orbits outside that zero set.Source: §2 · pp. 21–22

Ideal operations

• The orbit-indexed ideals satisfy J_S ∩ J_T=J_{S∩T}, J_S+J_T=J_{S∪T}, and ann_R(J_S)=J_{S^c}.The annihilator identity follows from orthogonality of the spectral idempotents.Source: §2 · pp. 22–23

Dimension modularity

• The dimension set function is modular: dim_Fq(I_T)+dim_Fq(I_U)=dim_Fq(I_{T∪U})+dim_Fq(I_{T∩U}).This follows by expressing T and U as disjoint unions of their differences and intersection.Source: §2 · p. 23

Multiplier automorphisms

• For gcd(a,ℓ)=gcd(b,m)=1, the multiplier μ_{a,b}(i,j)=(ai mod ℓ,bj mod m) is a bijection and is a code automorphism exactly when it preserves the defining set.Because multiplication by a and b commutes with q-Frobenius multiplication, μ_{a,b} permutes Frobenius orbits and acts on primitive spectral idempotents accordingly.Source: §2 · p. 24

D. Distance Bounds

• The 2D BCH bound gives d(C) ≥ ∏_{k∈Γ} δk when the defining set contains the required unions of δk−1 consecutive full zero hypercolumns.For Γ={k}, this yields d(C)≥δk; for Γ={1,2}, the certificate is δ1δ2.Source: D. Distance Bounds · p. 25
• Example 59 obtains d(C1)≥3 from two full zero strips in one direction and d(C2)≥4 from one full zero strip in each direction.The ideals are generated by g1=1+x+x^2 and g2=(1+x)(1+y) in F2[x,y]/⟨x^3−1,y^3−1⟩.Source: D. Distance Bounds · p. 25
• A rectangular block of zeros is insufficient for the two-sided BCH bound, which requires the union of complete strips; in Example 59, this prevents the false estimate d(C1)≥6 when the true distance is 3.Theorem 58 requires every point in each relevant hypercolumn, with cyclic consecutiveness allowing wraparound.Source: D. Distance Bounds · p. 25
• Beyond the BCH construction, multivariate shift bounds give d(C)≥D whenever every nonzero codeword’s zero set contains a set supporting an independent set of size D.Related approaches include Camion’s abelian BCH bound, Jensen’s concatenated inner/outer cyclic construction, and later refinements.Source: D. Distance Bounds · pp. 25–26

V. BIVARIATE BICYCLE CODES

• For bivariate bicycle codes, inversion-induced permutation conjugation removes transposes up to CSS code equivalence, while preserving abstract code parameters but potentially changing geometric or hardware implementations.The equivalence uses qudit permutations, stabilizer relabeling, global Hadamard, and monomial column scaling.Source: §V · pp. 26–27
• In the coprime binary case, the BB dimension is k=2 deg g(z), with g(z)=gcd(a(z),b(z),z^ℓm−1), while nontrivial codes of the specified form require ℓm divisible by a Mersenne or listed outlier prime.For odd ℓ and even m, a ring isomorphism decomposes the analysis into irreducible x-components, with component generators determined by gcds.Source: §V · pp. 27–28
• When ℓ and m are even, Gröbner bases yield k=2|m|=2 dim_Fq span_Fq(m), where m is the set of standard monomials outside the leading-term ideal.This provides a dimension formula for the ideal generated by a, b, x^ℓ−1, and y^m−1.Source: §V · p. 28
• The homology is pure exactly when ⟨a⟩∩⟨b⟩=⟨ab⟩; principal codes then generate all logical operators from principal annihilator generators, and odd ℓ,m guarantee principality.Pure BB codes with univariate or semi-periodic a,b are principal.Source: §V · pp. 28–29
• In the semi-periodic case, g(y) explicitly generates the associated cyclic code, with parity-check polynomial gcd(ζ(y)^k′ − 1, y^m − 1).The construction assumes a(x, y) = x^k + ζ(y) with kk′ = ℓ and defines g(y) through g(y)¯χ(y) = y^m − 1.Source: §V · p. 29
• The annihilator of a is generated by P(x, y), and the argument similarly yields a generator Q(x, y) for the annihilator of b.Source: §V · p. 29
• Restricting logical operators to the supports of A and B yields the upper bound d(H) ≤ min{d(H′_L), d(H′_R)}, while further shortening makes a looser bound computable.The restricted spaces form subsets of the full logical operators, so their minimum weights bound the true quantum distance from above; the shortened construction must exclude Z stabilizers.Source: §V · p. 29
• The classical auxiliary code H_L enforces u ∈ ker(A) while projecting out the stabilizer space through E_Bu = 0.This removes components supported on im(B), which contain left-block Z stabilizers.Source: §V · p. 29
• Because A and B are classical two-dimensional cyclic codes, orbit idempotents corresponding to e_O allow their intersections to be tailored precisely.Source: §V · p. 29
• Maximizing the kernel intersection by setting A = B can reduce the distance to two, so the dimension formula k = 2 dim(ker A ∩ ker B) does not determine distance alone.The role of orbits outside the intersection is therefore essential to the distance analysis.Source: §V · p. 29

VI. BB CODES FROM q-FROBENIUS ORBITS

• Bivariate bicycle codes are represented over R = F_q[x, y]/⟨x^ℓ−1, y^m−1⟩, with multiplication operators A and B defining the stabilizer structure.Source: §VI · p. 30
• The X-stabilizer code is CX = {(u, v) ∈ R^2 | au + bv = 0} = ker ΨX, with im ΨX = ⟨a, b⟩.Source: §VI · p. 30
• The kernels ker A and ker B are annihilator ideals, so BB codes become two-block index-2 quasi-cyclic codes whose constituents are two-dimensional cyclic codes.Source: §VI · p. 30
• Projecting au + bv = 0 onto each Frobenius orbit produces local constituent equations.Source: §VI · p. 30

1. Coupled zero-set region

• The active supports are S_a = {O ∈ Ω | a_O ≠ 0} and S_b = {O ∈ Ω | b_O ≠ 0}, partitioning the orbit contributions of a and b.Source: §VI · p. 30
• The active-support and zero-set descriptions use hatted regions for nonzero Wedderburn/Fourier supports and unhatted regions for zero sets on the same root grid.Frobenius-closed root-grid sets may be identified with their orbit subsets, while cardinalities count root-grid points rather than orbits.Source: §VI · p. 30

3. Free zero-set region

• Lemma 80 relates active-support and zero-set regions by complementing spectral supports, with bFa,b corresponding to Ta,b and analogous identities for the other regions.The proof follows from taking set-theoretic complements in the ambient root space and applying De Morgan’s laws.Source: §3 · p. 31

A. Dimension

• Theorem 81 identifies the code dimension with the common-zero region Ta,b, while Theorem 85 shows every orbit outside the free active-support region has trivial local homology.Consequently, every logical class has a representative supported strictly inside bFa,b, so uncoupled orbit components contribute no logical operators.Source: A. Dimension · pp. 31–32

B. Minimum Distance

• Theorem 88 bounds quantum distance by min{Ea, Eb, Na,b}, showing that mixed-block colon-ideal distances must be controlled alongside annihilator distances.The bound applies to both X- and Z-distances, and does not require semisimplicity of R.Source: B. Minimum Distance · pp. 33–34
• Theorem 90 refines the mixed-block term by excluding stabilizers from either block, yielding dZ ≥min{Ea, Eb, N ∗a,b}.The alternating exclusions prevent sparse check polynomials from capping the mixed-block distance through the stabilizer vector (b, −a).Source: B. Minimum Distance · pp. 34–35
• In the semisimple spectral description, ann⟨a⟩ and ann⟨b⟩ become cyclic-code regions, while ⟨b : a⟩ and ⟨a : b⟩ become uncoupled regions, producing computable distance terms.The resulting mixed-block term is Na,b = dR(C(Ub,a)) + dR(C(Ua,b)); omitting stabilizer subtraction gives a weaker but directly computable bound.Source: B. Minimum Distance · p. 35
• The refined mixed-block distance bound is computable from minimum Hamming weights of classical 2D cyclic codes via alternating stabilizer exclusions.Under Theorem 91, colon ideals and stabilizer ideals correspond to classical 2D cyclic codes defined by Frobenius-orbit zero sets.Source: B. Minimum Distance · p. 36
• For dense check polynomials, the un-excluded colon sum can already certify the code distance, whereas sparse weight-3 presentations may require stabilizer subcode subtraction to certify d > 6.The sparse-trinomial un-excluded bound is at most 6, while excluding the check-polynomial subcodes removes that ceiling.Source: B. Minimum Distance · p. 36
• Theorem 91 certifies d ≥23 for the explicit coprime code with parameters [[510, 104, d ≥23]].BCH runs give bounds 30, 23, and 24 for the component and mixed-block terms; inversion preserves these bounds for X-distance.Source: B. Minimum Distance · p. 37
• In the [[42, 2]] example, alternating stabilizer exclusion raises the refined mixed-block term from 4 to 5 and tightens the guaranteed Z-distance from 4 to 5.The true minimum distance is dZ = 7, while the BCH-style bound can be loose.Source: B. Minimum Distance · p. 37
• The BCH bound is used for simplicity but is often loose, so numerical evaluation of other bounds may yield better estimates.This limitation qualifies the example-based distance certificates.Source: B. Minimum Distance · p. 38

Sparse-generator design

• Sparse metachecks need not exist as single generators, so the construction may instead use multiple sparse generators of ann⟨a,b⟩ and treats their discovery as a design problem.In semisimple R, ann⟨a,b⟩ is principal, but its generator can be a dense central idempotent; two or more sparse generators may ease implementation.Source: C. Metachecks · p. 38

Construction strategy

• A one-dimensional construction starts from a cyclic code with generator g, chooses coprime codewords u and v, and searches for sparse metachecks in ann⟨g⟩, equivalently low-weight dual codewords.Extending this search to two dimensions is harder because R is no longer a Euclidean domain.Source: C. Metachecks · p. 38

Exact metacheck construction

• Exact metacheck generators satisfy ker(MX)=im(HX) and analogously on the Z side, so the operators capture all stabilizer redundancies.The construction uses ⟨m1,X,m2,X⟩=ann⟨a,b⟩ and ⟨m1,Z,m2,Z⟩=ann⟨ι(a),ι(b)⟩.Source: C. Metachecks · p. 38
• rank MX = rank MZ = k/2, while dMX ≤ min{wt(a), wt(b)} and dMZ ≤ min{wt(ι(a)), wt(ι(b))}.The rank result follows from the syndrome ideal’s dimension and rank-nullity; the distance bounds use a and b as codewords in the corresponding kernels.Source: C. Metachecks · p. 38

Explicit example

• dMX = 8 and dMZ = 8 for the R = F2[z]/⟨z15−1⟩ example, while the resulting BB code has dimension 22.Using the [11] [3] Hamming code, mX is its generator and the dual Simplex code gives dMX=8; the Z construction similarly gives dMZ=8.Source: C. Metachecks · p. 39

Automorphism groups

• Zero sets alone do not determine the stabilizer coordinate-permutation group, so HX and HZ are modeled as index-2 quasi-cyclic modules inside R2.The check modules have length 2ℓm, whereas the associated syndrome ideals are cyclic codes of length ℓm.Source: §VII · pp. 39–40

Local-Clifford automorphisms

• For prime-field qudit stabilizer codes, local-Clifford automorphisms combine coordinate permutations with single-qudit Clifford symmetries through Cl1 ≀ Sn.For CSS codes, pure permutations preserve X- and Z-check spaces separately; for p > 2, coordinate scalings enlarge this subgroup, while Hadamard-type exchanges and entangling Cliffords are treated separately.Source: §VII · pp. 39–40
• A physical automorphism U is characterized by an invertible check-mixing matrix satisfying the induced symplectic transformation condition on the full check matrix.The physical transformation acts as v 7→vMV, with MV ∈ Sp2n(Fq).Source: §VII · p. 40

Logical action

• Every local-Clifford automorphism induces a logical symplectic map, and it implements a nontrivial logical Clifford gate exactly when Mlog ≠ Id.Using a normalized symplectic logical basis, multiplying by JnL⊤ removes stabilizer-row contributions and yields the explicit logical action.Source: §VII · p. 40

Spectral constituents

• The X-check module decomposes into Frobenius-orbit constituents that are zero, coordinate lines, or a slope-defined graph across free, uncoupled, and coupled support regions.The spectral support combines the coupled and uncoupled active-support regions, while coupled constituents additionally retain slope information; the same framework applies to RZ.Source: §VII · p. 40

A. Permutation Automorphisms

• Translations by Z_ℓ×Z_m always act as permutation automorphisms because group-algebra multiplication commutes with every ring element.Source: A. Permutation Automorphisms · p. 41
• A monomial isometry preserving ker H_X and ker H_Z preserves both stabilizer row spaces and therefore the full CSS stabilizer.Source: A. Permutation Automorphisms · p. 41
• Multiplier and block-swap automorphisms require zero-set preservation plus exact slope matching on every coupled active-support orbit.Preserving the zero sets alone is necessary but insufficient; the slope condition depends on the actual elements a and b.Source: A. Permutation Automorphisms · pp. 41–42
• With no shifts or block swap, a multiplier is an automorphism exactly when it stabilizes both zero sets and satisfies µ_s(a)b = aµ_s(b).Source: A. Permutation Automorphisms · p. 42
• The coupled code’s symmetry group can be much smaller than the symmetry groups of its individual cyclic blocks because surviving multipliers must satisfy the additional slope test.Example 116 has a Frobenius multiplier group stabilizing both zero sets, but no nontrivial multiplier survives the slope test.Source: A. Permutation Automorphisms · pp. 42–43

B. ZX-Dualities & Fourier-Type Gates

• For every bivariate bicycle code, the signed block-swapping inversion D_0 exchanges the X- and Z-check modules, making it a ZX-duality and F D_0 a Fourier-type Clifford automorphism.Source: B. ZX-Dualities & Fourier-Type Gates · p. 43
• The coordinate duality D_0 has order four for p > 2 and order two over F_2, whereas its symplectic lift F D_0 is always an involution.Source: B. ZX-Dualities & Fourier-Type Gates · pp. 43–44
• The canonical subgroup ⟨Aut_perm(Q), F D_0⟩ is always present, but its logical action may be trivial and it need not exhaust the full local-Clifford automorphism group.Computations of find full groups strictly larger than the block-monomial part for several BB codes discovered in [3].Source: B. ZX-Dualities & Fourier-Type Gates · p. 44

C. Phase-Type & CZ-Type Clifford Gates

• A phase-type gate SD is symplectic exactly when D is symmetric, and it preserves the CSS stabilizer group exactly when rowspace(HXD) = rowspace(HZ) in the equal-rank setting.For symmetric D, SD is implemented by CZ-type gates off the diagonal and single-qudit phase gates on the diagonal [4, §2.4.2]; the stabilizer condition begins as rowspace(HXD) ⊆ rowspace(HZ).Source: C. Phase-Type & CZ-Type Clifford Gates · p. 44
• Over F2, the canonical duality D0 yields a valid CZ-type phase automorphism, whereas bare transversal phase gates require self-orthogonal X-stabilizers and are therefore highly constrained.For the example code, SD0 is valid while S⊗254 is obstructed by rowspace(HX) ≠ rowspace(HZ); over Fq, duality does not automatically supply a phase-type gate, and any suitable symmetric map depends on the individual code.Source: C. Phase-Type & CZ-Type Clifford Gates · pp. 44–45

VIII. LIFTS & PROJECTIONS

• A lifted p-cyclotomic coset modulo hn is p-invariant and partitions into cover cosets whose structure is governed by p-cyclotomic orbits modulo h, with special formulas for coprime and divisible cases.When gcd(h,n)=1, componentwise Chinese-remainder dynamics give coset lengths lcm(|C(h)_r|, κ); when h divides n and h divides n′, the lift splits into δ = gcd(c,h) cosets of size κh/δ.Source: §VIII · pp. 46–47
• In the general case h divides n, lifted cosets correspond to cycles of the affine map T(j) = λj + τ on Z_h, and a cycle of length d produces a cover coset of size κd.The cycle length is the least t > 0 satisfying S_t ≡ 0 (mod m_j); the simpler translation law applies when h divides n′, while omitted cases require the affine formulation.Source: §VIII · pp. 46–47
• Projection maps every cover coset exactly onto its base coset, with cover size K = wκ and a uniform w-to-1 orbit covering determined by the lift-index cycle length.The wrapping multiplicity w is the cycle length of T in the general divisible case; for h | n′, each base lift fiber splits into δ cover cosets.Source: §VIII · p. 47
• Theorem 123 decomposes each lifted base coset into disjoint cover cosets whose wrapping multiplicities sum to h, while projection is additive over the fiber and returns the base orbit sum with total weight h.The extreme cases are one fully wrapped coset with multiplicity h or h unwrapped copies with multiplicity 1.Source: §VIII · pp. 48–49
• Idempotent projections do not obey the orbit-sum scaling formula: ρ maps primitive idempotents to sums of primitive idempotents, not generally to w eO′.The orbit-sum identity concerns orbit sums rather than idempotents, and w eO′ is idempotent only when w ≡ 0 or 1 (mod p).Source: §VIII · p. 49
• A primitive cover idempotent survives projection if and only if its Frobenius-orbit frequencies are all divisible by h; otherwise the projection annihilates the spectral component.Surviving cover idempotents correspond bijectively to base primitive idempotents under frequency division by h, making the projection surjective on the supported spectral subspace.Source: §VIII · pp. 49–50
• A cover orbit sum projects as ρ(σu) = wσs and vanishes exactly when p divides w, with orbit length satisfying K = wκ.For h | n′, the fiber has δ = gcd(c,h) cover cosets of equal size K = κ(h/δ), giving uniform wrapping multiplicity w = h/δ.Source: §VIII · pp. 48, 50
• Together, Propositions 122 and 125 explain projected logical-operator vanishing through two distinct mechanisms: p-divisibility of orbit-sum wrapping multiplicities and h-divisibility of idempotent spectral support.The orbit-sum criterion is coefficient-based, whereas the idempotent criterion is spectral.Source: §VIII · p. 50

A. Logical Dimension of Covering Codes

• The proof uses canonical reduction between enlarged and base quotient rings, establishes ideal surjectivity, and derives the result through CSS dimension identities.This commutative-algebra approach independently reproves the covering lower bound and does not require semisimplicity.Source: A. Logical Dimension of Covering Codes · pp. 50–51
• For every covering degree h and field characteristic p, the logical dimension satisfies k_h ≥ k via a surjective quotient map and its kernel K_new.The canonical map ˜R/˜I → R/I induces a short exact sequence, yielding the dimension comparison.Source: A. Logical Dimension of Covering Codes · p. 50
• The cover preserves logical dimension exactly if and only if x^ℓ−1 and y^m−1 belong to the covering check ideal ⟨˜a,˜b⟩_˜R.Equivalently, ker ρ ⊆ ˜I, so the computable criterion distinguishes k_h = k from k_h > k.Source: A. Logical Dimension of Covering Codes · p. 51
• In the semisimple setting, the dimension formula becomes k_h = k + 2|∆Z|, where ∆Z consists of new common roots of ˜a and ˜b outside the base grid.The kernel is supported on the additional common spectral zeros and decomposes into Frobenius-orbit field components.Source: A. Logical Dimension of Covering Codes · pp. 51–52

Spectral restriction theorem

• For gcd(p, sℓm) = 1, the cover zero sets intersect the embedded subgroup G_s exactly as the injected base zero sets: Z_ea ∩ G_s = ϕ(Z_a) and Z_eb ∩ G_s = ϕ(Z_b).The corresponding T, U, and F regions also restrict via ϕ, while the covering condition imposes no restriction outside G_s.Source: B. Distance Certificates for Covering… · p. 52

Certificate limitations

• The injection ϕ(i, j) = (si, sj) does not preserve contiguous BCH certificates because inherited roots acquire s − 1 gaps.Cover distance bounds require control of all three terms, and preservation occurs only when new cover roots fill the induced gaps into consecutive full zero strips.Source: B. Distance Certificates for Covering… · p. 52
• Uniform covers expose a structural tradeoff: prescribed consecutive zero strips can yield high certified distance, but their inverse Fourier transforms produce dense checks incompatible with useful sparse BB codes.The paper notes check weights of 53 and 54 in an example requiring dense zero sets.Source: B. Distance Certificates for Covering… · p. 52

Illustrative example

• In the 3-fold lift of C = ⟨1 + x + x^2⟩, the base distance is 3 but the cover distance is exactly 2 because inherited strips occur at nonconsecutive indices.The cover contains 1 + x^3 of weight 2, while its nonempty defining set excludes weight-one codewords.Source: B. Distance Certificates for Covering… · pp. 52–53
• The classical-ideal example demonstrates that an individual distance can decrease under lifting, without proving that the full BB colon-ideal bound decreases.The full bound depends jointly on the certificates for E_a, E_b, and N_a,b.Source: B. Distance Certificates for Covering… · p. 53

IX. EXAMPLES

• The [[90, 8, ≤10]] code [9] yields n = 90, k = 8, and guaranteed minimum-distance lower bound 4 from Ea = Eb = 10 and Na,b = 4.The common-zero region has size 4, matching the derived logical dimension k = 8 through the framework’s formulas.Source: §IX · p. 53
• The [[162, 24, ≤6]] code [9] yields n = 162, k = 24, and guaranteed lower bound 4 from single-block distances 6 and mixed-block bound 4.Its common-zero region has size 12, while the colon-ideal term controls the final guarantee.Source: §IX · p. 53
• The [[434, 10, ≤26]] example [35] has n = 434, k = 10, and a colon-ideal minimum-distance bound of 2 despite single-block bounds of 112.Here Za = Zb, the generated ideals coincide through a unit multiple, and the analysis proves d ≥3 by excluding weight-two kernel vectors.Source: §IX · p. 53
• The paper constructs low-weight mixed-block logicals by choosing a unit c that acts identically on common-zero idempotents and combining cu with annihilator codewords.For the extreme example, the resulting unit has Hamming weight 97, and a mixed-block logical has the form (u, cu + f).Source: §IX · pp. 53–54

X. CONCLUSION

• The framework provides root-counting dimension formulas, colon-ideal minimum-distance bounds, automorphism and ZX-duality descriptions, and transformation rules under lifts and projections.These results arise from Wedderburn decomposition into finite-field components indexed by p-Frobenius orbits.Source: §X · p. 54
• The framework extends with little modification to finite fields Fq, while additive Fp-linearity makes stabilizer automorphism classification require additional care.The authors suggest extending BB codes to their linear closure as a possible research direction.Source: §X · p. 54
• Open directions include non-semisimple extensions with zero multiplicities and nilpotent local Artinian factors, twisted-torus constructions [21], and codes with additional blocks [18].The non-semisimple analysis is expected to be less computationally friendly because the simple field structure is lost.Source: §X · p. 54
• Future work targets stabilizer-weight reduction, orbit-grouping searches over semisimple grid sizes, and improved understanding of how BB structure influences decoding.The paper notes that example stabilizer weights may exceed the capabilities of most near-term technologies, while photonic platforms support arbitrary qubit connectivity.Source: §X · p. 54

BCH product-code construction

• Product codes have dimension (dim Cx)(dim Cy) and minimum distance dRx(Cx)dRy(Cy), giving the BCH lower bound dR(C) ≥ δxδy.The product zero set is a union of horizontal and vertical strips, and the distance identity follows from row and column weight arguments.Source: Appendix A · p. 55

Example 138

• In the ℓ=7, m=15 binary construction, strong and weak constituent codes give Ea,Eb ≥ 8 and mixed-block distance 3+3, hence dZ ≥ 6.The strong codes have designed and actual distances δx=δy=4, while the weak codes have γx=γy=2; the resulting BB code has equal X and Z distances.Source: Appendix A · p. 56
• The example’s 30 slack spectral points form three Frobenius orbits, which can be selectively included or excluded to adjust logical dimension while testing distance bounds.The orbit sizes are 12, 6, and 12, and the colon-ideal lower bound provides a fast distance check for resulting codes.Source: Appendix A · p. 57

Unit-multiple constructions

• When gcd(p,ℓm)=1, equal annihilator ideals imply equal logical dimensions and identical colon-ideal minimum-distance bounds for two BB codes.Distinct unit multiples can therefore produce potentially different codes while preserving n, k, and the same lower-bounded distance guarantees.Source: Appendix A · p. 57

Example 141

• For the [[90, 8, 10]] code of [3] over R = F2[x,y]/⟨x15−1,y3−1⟩, the following generator pairs define codes with parameters [[90, 8]] and the same colon-ideal distance lower bound.Source: Appendix A · p. 57