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Spectral Theory of Semisimple Bivariate Bicycle Codes

Eric Sabo, Mahir Bilen Can, David Marquis

arXiv:2608.27565v1quant-phcs.IT

TL;DR

Bivariate bicycle codes have often been found through numerical searches, leaving their algebraic design principles and guaranteed parameters unclear. This paper develops a semisimple Frobenius-orbit framework with idempotents, distance bounds, and symmetry analysis, and illustrates deterministic construction and explicit examples.

  • Problem

    Good bivariate bicycle codes have largely been discovered numerically, leaving deterministic design with first-principles guarantees for dimension and distance limited.

  • Method

    The paper decomposes the semisimple bivariate quotient ring into Frobenius-orbit field components and analyzes code parameters, mixed-block distance bounds, and symmetries through those components.

  • Results

    The framework derives a root-counting dimension formula, a colon-ideal lower bound for quantum distance, structured automorphism and ZX-duality descriptions, and exact lift/projection dimension formulas.

  • Takeaways & Limitations

    The spectral approach enables deterministic analysis and enumeration of bivariate bicycle codes without relying on numerical searches or explicitly writing stabilizers.

Abstract

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Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.

Orbits, idempotents, and regions

The paper places bivariate bicycle codes in a semisimple two-dimensional cyclic-code framework, using Frobenius-orbit components to analyze dimensions, distances, symmetries, and lifts.

  • Spectral framework: In the semisimple regime, the ring decomposes into finite-field components indexed by q-Frobenius orbits, making polynomial projections central to code analysis.The regime assumes gcd(ℓm, q) = 1.
  • Logical dimensions: The logical dimension is determined by the common-zero region, while orbits where only one defining polynomial vanishes are homologically trivial.The X- and Z-logical spaces are related by an exact algebraic duality via the canonical coordinate involution.
  • Minimum distance: A colon-ideal bound captures mixed-block logical operators and yields a sharper computable lower bound on quantum distance after alternating stabilizer exclusion.This separates roots controlling logical dimension from those controlling distance.
  • Symmetries: Coordinate-permutation automorphisms depend on spectral slopes on the coupled active-support region, not only on zero sets.The framework also establishes signed block-swapping ZX-duality, characterizes phase-type CZ gates, and describes metachecks.
  • Lifts and construction: The approach analyzes code lifts and projections through check-ideal algebra and gives an exact dimension formula based on strictly new common roots over finite fields.The paper emphasizes deterministic analysis and enumeration rather than numerical searches or stabilizer construction.

III. 1D CYCLIC CODES

The one-dimensional cyclic-code review develops the polynomial, ideal, defining-set, and Frobenius-coset tools later used in the bivariate theory, including distance guarantees from BCH conditions.

  • Algebraic representation: Cyclic codes of length n correspond bijectively to ideals of Fq[x]/⟨x^n−1⟩, with generator and check polynomials determining complementary code structures.If deg g(x) = r and deg h(x) = k, then k = n − r and the code has parameters [n, k]q.
  • Roots and defining sets: Frobenius conjugates group roots into q-cyclotomic cosets, which determine irreducible factors and defining sets over Fq.The condition gcd(q, n) = 1 ensures distinct roots in the factorization.
  • Duality: The dual of a cyclic code is cyclic, with its defining set obtained by negating the complement of the original defining set modulo n.The generator of the dual uses the reciprocal check polynomial, normalized by h(0).
  • Automorphisms: Multiplier automorphisms act by c(x) ↦ c(x^s) and preserve a cyclic code exactly when they preserve its defining set.Because they permute coordinates, they preserve Hamming weight.
  • Distance bounds: A BCH defining set containing δ − 1 consecutive exponents guarantees minimum distance d(C) ≥ δ.This constructs cyclic codes with prescribed distance through a defining-set condition.

IV. 2D CYCLIC CODES

Two-dimensional cyclic codes are ideals in a bivariate quotient ring, but their genuinely two-dimensional structure depends on the relation between ℓ and m and becomes spectrally tractable in the semisimple case.

  • Definition and structure: A 2D cyclic code is an ideal in R = Fq[x, y]/⟨x^ℓ−1, y^m−1⟩, where the non-Euclidean polynomial ring generally requires multiple generators.Existing approaches use canonical generators and polynomial algorithms such as Gröbner bases.
  • One-dimensional collapse: When gcd(ℓ, m) = 1, the coordinate group and ring reduce to one-dimensional cyclic structures of length ℓm.A ring homomorphism maps x to z^m and y to z^ℓ and is surjective.
  • Semisimple regime: A genuinely two-dimensional theory emerges when gcd(ℓ, m) ≠ 1, while semisimplicity requires gcd(ℓm, q) = 1.The semisimple condition is equivalent to the characteristic dividing neither ℓ nor m.

A. Orbits

The orbit theory partitions Zℓ × Zm into q-Frobenius strata whose sizes and counts are computable from multiplicative orders, supporting the finite-field decomposition underlying the spectral framework.

  • Orbit decomposition: A q-Frobenius orbit of (i, j) is the two-dimensional analogue of a cyclotomic coset, and each orbit corresponds to a minimal idempotent component in the semisimple ring.Orbit length determines the degree of the associated field extension.
  • Stratified counting: For strata Λd,e, every orbit has length L(d, e) = lcm(ord_d(q), ord_e(q)), and the number of contained orbits is ϕ(d)ϕ(e)/L(d, e).The multiset of orbit sizes is determined by the multiplicative orders for divisors of ℓ and m.
  • Burnside and inversion: The fixed-point count of the tth Frobenius power factors as gcd(ℓ, q^t−1)gcd(m, q^t−1), enabling Burnside counting of all orbits.Möbius inversion further recovers the number of points with each exact orbit length.
  • Example: For q = 2 and ℓ = m = 3, the grid has exactly 5 Frobenius orbits: one of length 1 and four of length 2.Both the stratum calculation and Burnside/Möbius calculation give this result.

B. Idempotents

The paper constructs primitive and Frobenius-orbit idempotents through evaluation, interpolation, and Fourier formulas, then uses them to decompose the semisimple ring into finite-field components.

  • Primitive idempotents: The evaluation map identifies the scalar-extended ring RK with K^ℓm, where primitive idempotents correspond to coordinate vectors.Lagrange interpolation and discrete Fourier transform give two explicit formulas for these idempotents.
  • Frobenius descent: q-Frobenius permutes primitive idempotents along orbits, while each orbit sum is Frobenius-fixed and therefore lies in R.Orbit idempotents are pairwise orthogonal and sum to 1.
  • Example: The paper illustrates the construction for q = 2 and ℓ = m = 3, where explicit orbit-idempotent polynomials descend to F2.The example lists orbit sums such as eO0,0, eOx, eOy, eO+, and eO−.
  • Wedderburn decomposition: The orbit idempotents induce a product decomposition of R into components e_OR, each isomorphic to the finite field Fq^|O|.The orbit size is the extension degree of the corresponding Wedderburn component.
  • Dimensions: As an Fq-module, each component e_OR has dimension |O|, and direct-sum dimensions add over selected orbit sets.This yields dimension formulas for ideals assembled from orbit components.

C. Lattice Theory

Selected subsets of Frobenius orbits correspond to selected idempotents and ideals, and every ideal has a unique representation of this form.

  • Selected ideals: For a subset S ⊆ Ω, the S-selected idempotent is eS = ΣO∈S eO and the corresponding ideal is JS = eSR.These constructions select precisely the orbit components indexed by S.
  • Bijection: Theorem 54 establishes a bijection between subsets of Ω and ideals of R, with every ideal J ◁ R uniquely equal to JS.The result also identifies the corresponding orbit-based structure after evaluation.

2. Under this correspondence,

The orbit correspondence equips ideals with set-theoretic lattice operations, dimension modularity, and multiplier symmetries expressed through defining sets.

  • Ideal lattice: The number of ideals of R is 2^|Ω|, because each ideal corresponds uniquely to a subset of Frobenius orbits.The subset-to-ideal correspondence turns orbit selection into ideal enumeration.
  • Zero sets: Under evaluation, an ideal associated with a zero set T consists exactly of functions vanishing at every point of T.This identifies ideal membership with coordinatewise vanishing in K^ℓm.
  • Orbit correspondence: Frobenius closure makes every zero set a union of q-Frobenius orbits, yielding the unique orbit subset that represents the ideal.The equality J = I(Z(J)) connects an ideal to its zero set and selected orbit idempotent.
  • Lattice operations: Ideal intersections and sums correspond to intersections and unions of selected orbit sets, while annihilation corresponds to the complementary orbit set.In particular, annR(JS) = JSc.
  • Modularity: The set function f(T) = dimFq(IT) satisfies modularity: dimFq(IT) + dimFq(IU) = dimFq(IT∪U) + dimFq(IT∩U).This follows from the product structure of the evaluated ring and additivity over disjoint unions.
  • Symmetries: Multiplier maps µa,b(i,j) = (ai mod ℓ, bj mod m), with gcd(a,ℓ) = gcd(b,m) = 1, permute Frobenius orbits and preserve a code exactly when they preserve its defining set.The q-Frobenius map is the special diagonal multiplier in the semisimple case.

D. Distance Bounds

The paper develops multivariate BCH-style distance certificates based on full zero strips, demonstrates one- and two-sided bounds, and emphasizes that rectangular zero blocks alone are insufficient.

  • 2D BCH bound: The 2D BCH bound uses consecutive full zero hypercolumns in one or both coordinate directions to certify minimum distance.For one coordinate, δk − 1 consecutive full hypercolumns give d(C) ≥ δk; multiple directions yield a product certificate.
  • Examples: For C1, the zero set {1,2} × Z3 gives the one-sided certificate d(C1) ≥ 3.The generator is g1 = 1 + x + x^2 in F2[x,y]/⟨x^3 − 1,y^3 − 1⟩.
  • Examples: For C2, the zero set ({0} × Z3) ∪ (Z3 × {0}) gives the two-sided certificate d(C2) ≥ 4.The bound is 2 · 2 = 4 for g2 = (1 + x)(1 + y).
  • Applicability caveat: A rectangular block of zeros does not suffice for the product bound; the defining set must contain the required union of full strips.For C1, the rectangle would incorrectly predict d(C1) ≥ 6, while the true distance is 3.
  • Applicability caveat: The theorem requires cyclic consecutiveness and membership of every point in each required hypercolumn, not merely a visually consecutive rectangle.The index sets may wrap around modulo the relevant code length.
  • Extensions: More general multivariate bounds use independent-set constructions, where an independent set of size D implies d(C) ≥ D.The text also situates the result among generalized apparent-distance, Hartmann–Tzeng, Roos, and shift bounds.
  • BB-code structure: The BB-code framework further supplies algebraic dimension formulas and structural results, including k = 2 deg g(z), Gröbner-basis dimension counts, and principal-ideal criteria.These results complement distance analysis with explicit parameter and ideal-structure calculations.

VI. BB CODES FROM q-FROBENIUS ORBITS

The section represents bivariate bicycle codes over a semisimple quotient ring and decomposes their structure using q-Frobenius orbit constituents. The resulting framework identifies parity-check kernels, ideals, and local constituent equations.

  • Code construction: Bivariate bicycle codes use polynomials a,b in R = Fq[x,y]/⟨x^ℓ−1,y^m−1⟩ over a prime-power field.The code is represented in the standard monomial basis, with multiplication operators A and B.
  • Check modules: The X-check code is CX = {(u,v) ∈ R2 | au+bv = 0}, equivalently the kernel of ΨX.The image of ΨX is the ideal ⟨a,b⟩.
  • Check modules: The code decomposes into two-block quasi-cyclic constituents whose components are two-dimensional cyclic codes.The annihilators ker A and ker B are ideals in R.
  • Frobenius-orbit decomposition: Projecting au+bv = 0 onto a Frobenius orbit O yields a local constituent equation involving the projected check polynomials.The active supports Sa and Sb collect orbits where aO and bO are nonzero.
  • Frobenius-orbit decomposition: Active-support and zero-set versions partition the orbit space according to whether each defining polynomial vanishes.Hatted regions use Fourier supports, while unhatted regions use zero sets on the root grid.

2. Uncoupled zero-set regions

The section shows that logical dimension and homology are controlled by common-zero or free-support regions, while uncoupled orbit components are homologically trivial. It also develops distance bounds that explicitly account for mixed-block logical operators.

  • Region duality: The active-support and zero-set partitions are Fourier-dual: both-nonzero regions correspond to neither-zero regions, and mixed regions exchange labels.This follows by taking spectral supports as complements of zero sets and applying De Morgan’s laws.
  • Dimension: The logical dimension is determined by the common-zero region Ta,b, where both defining polynomials vanish.The annihilator ann⟨a,b⟩ is supported exactly on these orbits, giving k = 2 dimFq(ann⟨a,b⟩).
  • Homology: Orbits outside the free active-support region have trivial local homology, so uncoupled components do not contribute logical classes.Every logical class admits a representative supported within the free active-support region.
  • Homology: The canonical involution with block exchange gives a dimension-preserving algebraic duality between logical Z and X spaces.The map (u,v) 7→ (ι(v),−ι(u)) is a bijective automorphism between the logical spaces.
  • Minimum distance: dX,dZ ≥ min{Ea,Eb,Na,b}, and d(Q) ≥ min{Ea,Eb,Na,b}.The mixed-block term Na,b captures logical operators spanning both blocks, complementing the single-block terms Ea and Eb.
  • Minimum distance: Colon-ideal terms are necessary because large distances in ann⟨a⟩ and ann⟨b⟩ alone do not ensure a large quantum distance.Alternating stabilizer exclusions refine the mixed-block contribution into a computable bound from classical 2D cyclic-code distances.
  • Minimum distance: Theorem 91 translates annihilator and colon ideals into orbit-region classical-code distances, making the refined bound computable.The un-excluded colon sum can already certify distance when dense check polynomials make the stabilizer ceiling large.
  • Examples: 23 is the certified minimum distance for the BCH-based example with parameters [[510,104,d ≥23]].Inversion preserves the BCH-designed bounds, yielding dX ≥23 and d = min{dX,dZ} ≥23.

C. Metachecks

Metachecks encode stabilizer redundancies through annihilator ideals and can form an exact two-block chain complex. Explicit constructions demonstrate strong metacheck distances, while sparse generation remains a design challenge.

  • Metacheck construction: The metacheck ideals are ann⟨a,b⟩ for X and ann⟨ι(a),ι(b)⟩ for Z.These ideals define the valid syndrome spaces and their associated metacheck codes.
  • Limitations: Sparse metacheck generators are not guaranteed to exist, so constructing them is treated as a design problem.Using multiple sparse generators can ease the burden, but two-dimensional searches are harder because the ring is not a Euclidean domain.
  • Design perspective: Restricting metachecks to weight at most w corresponds roughly to an LRC locality r = w − 1 in an exact two-block group-algebra complex.This suggests using LRC results to design BB metachecks from first principles.
  • Exactness: The metacheck matrices have rank k/2 and capture all X- and Z-stabilizer redundancies.Their kernels coincide with the corresponding stabilizer images, producing exactness under the stated generation condition.
  • Distance bounds: The metacheck distances satisfy dMX ≤ min{wt(a),wt(b)} and dMZ ≤ min{wt(ι(a)),wt(ι(b))}.The bounds follow because a and b, or their involution images, lie in the relevant stabilizer images.
  • Example: For the binary [11] [3] Hamming-code construction, the BB code has dimension 22 and both metacheck distances equal 8.The X metacheck distance matches the upper bound from the constructor weights, and the Z construction also gives dMZ = 8.

VII. AUTOMORPHISMS

The automorphism analysis treats BB stabilizers as quasi-cyclic modules and shows that coordinate symmetries depend on both orbit supports and spectral slopes. Physical automorphisms induce explicit logical Clifford actions.

  • Spectral structure: Zero sets alone do not determine coordinate-permutation automorphisms of semisimple BB codes.The check modules must instead be viewed as index-2 quasi-cyclic submodules of R2.
  • Automorphism notions: Local-Clifford automorphisms are symplectic transformations preserving the stabilizer code, while permutation automorphisms preserve X- and Z-check row spaces separately.For q > 2, coordinate scalings enlarge pure permutation symmetries, and Hadamard-type operations may exchange the two sides.
  • Symmetry constraints: Multiplier maps require stabilizer and slope conditions and are not automatically automorphisms.Thus preserving a cyclic or zero-set structure is insufficient without matching the coupled-region slope data.
  • Logical action: A physical automorphism induces a logical symplectic map Mlog, which is nontrivial exactly when Mlog ≠ Id.A normalized logical basis removes stabilizer-row contributions and yields the explicit logical Clifford action.
  • Spectral slopes: On coupled active-support orbits, each check-module constituent is a line whose slope carries additional symmetry-relevant information.Uncoupled regions give coordinate lines, while the coupled region gives a graph determined by the relative check-polynomial ratio.

A. Permutation Automorphisms

The paper characterizes structured permutation and block-monomial automorphisms through kernel preservation, zero-set behavior, and exact slope matching across Frobenius-orbit constituents.

  • Translations: Diagonal translations by Z_ℓ × Z_m always act as permutation automorphisms of the code.Multiplication by monomials commutes with every element of the commutative group algebra.
  • Stabilizer criterion: A coordinate isometry preserves the CSS stabilizer when it preserves both ker H_X and ker H_Z.The proof uses rowspace(H) = ker(H)^⊥ and preservation of orthogonal complements.
  • Multiplier and block symmetries: Zero-set preservation is necessary but not sufficient: coupled active-support constituents must also satisfy exact slope matching after orbit relabeling and relative shifts.The slope condition depends on the actual elements a and b, not only on their zero sets.
  • Multiplier and block symmetries: Block-preserving maps require multiplier stabilization of both zero sets, while block-swapping maps require the multiplier to exchange them.These conditions are stated for ker H_X and must also hold for the Z-check generators to obtain a full code automorphism.
  • Multiplier and block symmetries: The strict multiplier stabilizer can be much smaller than either single-block symmetry group, including cases where only the identity survives the slope test.An explicit example has the Frobenius multiplier group stabilizing both zero sets but no nontrivial multiplier satisfying slope matching.
  • ZX-dualities: The signed block-swapping inversion D_0 exists for every bivariate bicycle code and yields a Fourier-type Clifford automorphism through its lift F D_0.D_0 maps the X-check module to the Z-check module, while the generated subgroup is always present; its logical action may still be trivial.

C. Phase-Type & CZ-Type Clifford Gates

The paper characterizes phase-type and CZ-type Clifford automorphisms through symplecticity and stabilizer preservation, while distinguishing folded dualities from bare transversal gates. Its lift-and-projection analysis further relates orbit coverings, wrapping multiplicities, and spectral-support survival.

  • Phase-type constructions: D ≠ Id generally produces nonlocal Clifford operations, while symmetric D yields CZ gates off the diagonal and single-qudit phase gates on the diagonal.This defines the phase-type fold-transversal construction for symmetric bivariate bicycle codes.
  • Symplectic and stabilizer conditions: D⊤ = D is necessary and sufficient for SD to be symplectic; stabilizer preservation requires rowspace(HXD) = rowspace(HZ) in the equal-rank setting.The stabilizer condition arises from mapping each X-stabilizer row (hX | 0) to (hX | hXD).
  • Field-dependent limitations: Over F2, the canonical duality D0 is simultaneously a symmetric involution and a valid phase-type map, but this coincidence fails over Fq.Over Fq, antisymmetry obstructs the symmetry test, while the unsigned swap fails the stabilizer condition because of the sign in HZ.
  • Folded duality: The folded Hadamard FD0 is always a valid Clifford automorphism, whereas the unpermuted transversal Hadamard is obstructed when the common-zero region is not inversion-invariant.For the example, −T ≠ T modulo 127, so the spectral supports of RX and RZ differ.
  • CZ-type versus bare gates: The entangling CZ-type phase operator SD0 is valid along D0's two-cycles, while bare transversal phase gates require rowspace(HX) = rowspace(HZ) and are therefore highly constrained.The same row-space equality characterizes bare phase preservation in the equal-rank setting and implies X-stabilizer self-orthogonality.
  • Lifts and projections: A cover coset has size K = w·κ, and projection acts as a uniform w-to-1 orbit covering; fiber wrapping multiplicities sum to h.The extreme cases are one fully wrapped coset with w1 = h or h unwrapped copies with wi = 1.

A. Logical Dimension of Covering Codes

The paper gives an algebraic theory of BB-code covers in which quotient-ring surjections yield an exact logical-dimension relation. Logical dimension never decreases, and in semisimple cases its increase is counted by new common spectral roots.

  • Cover construction: The covering construction enlarges the ring by replacing ℓ,m with uℓ,tm, with covering degree h = ut and check polynomials reducing to the base checks.The monomial exponents of the cover checks are obtained by adding multiples of ℓ and m to the base exponents.
  • Algebraic dimension formula: Theorem 126 constructs a surjective quotient-space map for an h-cover and identifies its kernel Knew through a short exact sequence.The construction applies to arbitrary covering degree and does not require semisimplicity.
  • Algebraic dimension formula: The exact dimension relation is kh = k + 2 dimFq(Knew), so newly introduced kernel components account for every added logical qudit.The base and covering dimensions arise from quotient spaces R/I and ˜R/˜I, respectively.
  • Algebraic dimension formula: The logical dimension of an h-cover BB code is never smaller than that of its base code for any h ≥1 and field characteristic p.This independently reproves the covering-code lower bound without restricting the characteristic or covering degree.
  • Equality criterion: The logical dimension remains unchanged exactly when x^ℓ−1 and y^m−1 belong to the covering check ideal ⟨˜a,˜b⟩˜R.Equivalently, the covering projection kernel is contained in the covering check ideal.
  • Spectral interpretation: In semisimple covers, the formula reduces to kh = k + 2|∆Z|, where ∆Z contains new common roots of ˜a and ˜b outside the embedded base grid.Frobenius-orbit field components supported on these strictly new common zeros form Knew.

B. Distance Certificates for Covering Codes

Covering projections preserve the base code’s spectral information on an embedded subgroup, but they do not generally preserve minimum-distance certificates. Consecutive-strip requirements can be disrupted by the lift, creating a tension between sparse checks and distance guarantees.

  • Spectral restriction: Under a semisimple uniform cover, the frequency-domain projection embeds base indices by ϕ(i,j) = (si,sj) into the subgroup Gs.The coefficient reduction map ρ corresponds to this injection in the frequency domain.
  • Spectral restriction: Theorem 130 preserves the base zero sets and derived regions exactly after restriction to Gs, including T, U, and F regions.Outside Gs, the covering condition imposes no restriction on the cover zero sets.
  • Certificate limitations: The restriction theorem does not preserve minimum-distance certificates because ϕ spaces inherited consecutive roots by s−1 gaps.Preservation requires strictly new cover roots to fill those gaps into consecutive full zero strips.
  • Structural tension: Forcing consecutive zero strips can produce dense spatial-domain checks, while useful BB codes require sparse checks with fewer degrees of freedom.The paper identifies this as a tension between low-density parity-check structure and apparent-distance certificates.
  • Example: In the 2D cyclic example, the base code has d(C) = 3, whereas its literal 3-fold lift has d( eC) = 2.The lifted zero strips occur at x-indices 3 and 6 rather than consecutively; 1+x^3 provides a weight-two codeword.
  • Certificate limitations: A decrease in one lifted classical ideal’s distance does not by itself imply a decrease in the BB colon-ideal bound, which depends on eEa, eEb, and eNa,b.All three distance terms must be controlled for a bound on the cover distance.

IX. EXAMPLES

The examples validate the spectral framework on existing and constructed bivariate bicycle codes, showing how logical parameters, distance bounds, and code families can be controlled algebraically. They also identify scope limits and directions for extending the framework.

  • Examples: Three literature examples recover code parameters [[90, 8, ≤10]], [[162, 24, ≤6]], and [[434, 10, ≤26]] from spectral data.The examples compute zero orbits, common-zero regions, dimensions, and distance-related quantities.
  • Examples: The [[434, 10, ≤26]] example has single-block bounds of 112 but a colon-ideal minimum-distance bound of only 2.The mixed-block term evaluates to 2, dominating the larger uncoupled bounds.
  • Examples: When Za = Zb but a ≠ b, a unit c exists with a = bc, so the generated ideals coincide without forcing distance two.The example proves d ≥3 by excluding weight-two kernel vectors and single-block weight-two logicals.
  • Examples: The common-zero idempotent eO constructs a unit c that acts identically on the common-zero region, producing an explicit multiplier of Hamming weight 97.This construction explains how mixed-block logical operators can arise when cu is close to a codeword of ann⟨b⟩.
  • Conclusion and outlook: The authors note that some stronger module-theoretic results remain unassessed and that stabilizer codes require care because they are Fp-linear rather than Fq-linear.These observations delimit the current commutative-ring presentation and its field-generality claims.
  • Conclusion and outlook: The framework preserves logical dimension and colon-ideal distance certificates under independent unit changes, while structured symmetry analysis imposes slope constraints on valid automorphisms.The conclusion also points to extensions involving broader group-algebra codes, twisted tori, non-semisimple rings, stabilizer weights, and decoding.

Appendix A: BCH-Based Constructions & Product Codes

The appendix constructs bivariate bicycle codes from one-dimensional cyclic constituents using product-code ideals and BCH distance guarantees. The resulting colon-ideal analysis provides explicit, computable lower bounds while unit multiples and slack orbits support controlled code variation.

  • Product-code construction: The zero set of a product generator is a union of horizontal and vertical strips determined by the one-dimensional zero sets.This follows from the multiplicative evaluation map: a product vanishes when either constituent vanishes.
  • Product-code construction: Product-code ideals Prod(Cx, Cy) are tensor products whose dimensions multiply and whose minimum distances equal the product of constituent distances.BCH design-distance bounds therefore lift directly to the two-dimensional product code.
  • Example 138: In Example 138, strong constituent codes have BCH design distances δx = δy = 4, while weak codes have γx = γy = 2.The strong zero sets contain three consecutive residues, and the weak binary length-7 codes are single-parity-check codes.
  • Example 138: The constituent products yield Ea and Eb lower bounds of 8, while the colon ideals each have distance 3.The mixed-block term is therefore Na,b = 3 + 3.
  • Example 138: The resulting BB code satisfies dZ ≥ min(8, 8, 3 + 3) = 6, with equal X and Z distances.The bound applies to any choice of stabilizer generator polynomials a and b for the prescribed annihilator ideals.
  • Design variation: Slack Frobenius orbits can be selectively assigned to Za and Zb to adjust logical dimension, after which colon-ideal bounds provide a fast distance screen.The example identifies 30 points in three slack orbits of sizes 12, 6, and 12.
  • Design variation: Replacing generators by elements with the same annihilators, or by distinct unit multiples, preserves logical dimension and the same colon-ideal distance bounds.Theorem 140 establishes this preservation in the semisimple case gcd(p, ℓm) = 1.
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