Lifted-Product QLDPC Codes in the Polynomial Domain

• Finite-length implementations require explicit algebraic descriptions, controlled block size, and structured hardware-oriented representations.Source: §I · p. 1
• Polynomial and quasi-cyclic descriptions lift small base matrices into large sparse binary parity-check matrices.The formulation uses the quotient ring F2[D]/(D^L + 1) for entrywise circulant lifting.Source: §I · p. 1
• The construction formulates lifted-product parity-check matrices directly over R_L = F2[D]/(D^L + 1).Small polynomial base matrices compactly describe the corresponding circulant-lifted binary matrices.Source: §I · p. 1
• Conjugate-compatible circulant lifting provides a polynomial-domain proof of the CSS commutation condition H_XH_Z^T = 0.Source: §I · p. 1
• The framework connects convolutional-code polynomial parity checks with finite-length lifted-product construction through tail-biting circulant lifting.It is illustrated with a constraint length 7, rate R = 1/2 NASA convolutional code and a 3 × 4 polynomial parity-check matrix.Source: §I · p. 1
• The quotient ring identifies polynomials modulo D^L + 1, giving unique representatives of degree less than L and reducing exponents modulo L.Because the field has characteristic two, D^L = 1 and D^-1 = D^(L-1).Source: A. Polynomial Rings and Quotient… · p. 2

• A circulant permutation matrix shifts the identity matrix row-by-row by s positions to the right modulo L.Source: B. Circulant Matrices and Lifting · p. 2
• This operation provides a finite circulant, or tail-biting, block representation of polynomial convolutional-code components.Source: B. Circulant Matrices and Lifting · p. 2
• The lifting operation maps p(D) in R_L to a binary circulant matrix by summing shift matrices over F2.Source: B. Circulant Matrices and Lifting · p. 2
• A coefficient a_i = 1 includes the circulant shift by i, while a_i = 0 omits it; the rows are cyclic shifts of the coefficient vector.Source: B. Circulant Matrices and Lifting · p. 2
• The polynomial–circulant correspondence replaces each polynomial entry with an L × L binary circulant matrix.Source: B. Circulant Matrices and Lifting · p. 2
• The resulting matrix has dimensions mL × nL and consists of L × L circulant blocks.Source: B. Circulant Matrices and Lifting · p. 2

• QBP performs message passing on CSS Tanner graphs while maintaining beliefs over the four Pauli symbols I, X, Y, and Z.Source: C. Depolarizing Channel Model and… · p. 2
• Under the depolarizing channel, each of X, Y, and Z occurs independently with probability p.Source: C. Depolarizing Channel Model and… · p. 2
• QBP accounts for coupling introduced by Y errors, then checks its hard Pauli estimate for syndrome consistency and logical correctness.Source: C. Depolarizing Channel Model and… · p. 2
• QBPGD interleaves short QBP runs with G reliability-based decimation steps that fix highly reliable qubits.This can break symmetric failures caused by short cycles and degeneracy, but may increase decoding cost.Source: C. Depolarizing Channel Model and… · p. 2
• RL-S retains BP message updates but replaces flooding with a learned sequential variable-node update order during inference.Its policy is trained offline and selects the next variable node from local state information.Source: C. Depolarizing Channel Model and… · p. 2

• Polynomial conjugation maps p(D) to p*(D) = p(D^-1) within the quotient ring, with D^L interpreted as 1.Source: A. Algebraic Properties · p. 3
• The conjugation map is F2-linear, involutive, and multiplicative.Source: B. Matrix Conjugate Operations · p. 3
• The conjugate transpose transposes a polynomial matrix and replaces each entry with its conjugate polynomial.Source: B. Matrix Conjugate Operations · p. 3
• The paper performs conjugation within R_L using its defined quotient-ring operation, distinct from Hermitian transpose terminology.Source: B. Matrix Conjugate Operations · p. 3
• For conformable polynomial matrices, conjugate transpose distributes over addition and reverses multiplication order.Source: B. Matrix Conjugate Operations · p. 3
• For Kronecker products, conjugation distributes as (A ⊗ B)* = A* ⊗ B*.Source: B. Matrix Conjugate Operations · p. 3

Lifting map

• The lifting map LL sends elements of R_L to binary L × L circulant matrices.The map is used entrywise to lift polynomial matrices.Source: A. Lifting map and the… · p. 3
• LL preserves addition and multiplication, including entrywise operations on lifted matrices.These identities follow because circulant-matrix arithmetic corresponds to polynomial arithmetic in R_L.Source: A. Lifting map and the… · p. 3

Conjugate compatibility

• Conjugate-compatible lifting requires LL(p∗) = LL(p)T for every p ∈ R_L.This condition identifies polynomial conjugation with matrix transposition after lifting.Source: A. Lifting map and the… · p. 3
• Standard circulant lifting satisfies LL(p∗) = LL(p)T for all p ∈ R_L and extends this identity to matrices.The proof checks monomials, uses cyclic-index reduction modulo L, and extends by F2-linearity and entrywise application.Source: A. Lifting map and the… · p. 3

Polynomial check matrices

• The construction allows arbitrary polynomial parity-check base matrices over R_L.These matrices are used to define the asymmetric lifted-product checks.Source: B. Lifted-product polynomial check… · p. 3
• The asymmetric LP polynomial check matrices use Kronecker products over R_L and identity matrices I_s.The Kronecker product is taken over the coefficient ring R_L.Source: B. Lifted-product polynomial check… · p. 3

Conditional LP orthogonality theorem

• Theorem 1 states that conjugate-compatible lifting makes lifted-product matrices satisfy the required orthogonality relation for any base matrices over R_L.The theorem is conditional on the lifting map satisfying the conjugate-compatibility definition.Source: C. Conditional LP orthogonality theorem · p. 3

NASA example

• The lifted pair satisfies HXHZT = 0 over F2 and therefore defines a valid CSS code.The example is intended to illustrate the polynomial-domain construction and conjugate-compatible lifting argument.Source: C. Conditional LP orthogonality theorem · pp. 3–4

Conditional LP orthogonality theorem

• The polynomial-domain proof expands the product into two Kronecker terms, applies the mixed-product rule, and cancels them because the coefficient ring has characteristic two.The proof uses commutativity of R_L and the identity (A ⊗ C)(B ⊗ D) = (AB) ⊗ (CD).Source: C. Conditional LP orthogonality theorem · p. 3
• Binary lifting transfers the polynomial orthogonality identity to HXHZT = 0.The transfer uses the lifting map's multiplicative property and conjugate compatibility.Source: C. Conditional LP orthogonality theorem · p. 4

NASA example

• The NASA example uses a constraint length 7, rate R = 1/2 convolutional code with a symmetric lifted-product construction.It sets B1 = B2 = B, m = 1, and n = 2.Source: C. Conditional LP orthogonality theorem · p. 4
• The example forms polynomial LP checks, applies standard circulant lifting entrywise, and obtains binary matrices with 35 columns and 14 rows.The dimensions are HX, HZ ∈ F2^14×35.Source: C. Conditional LP orthogonality theorem · p. 4

Code construction

• The selected 3 × 4 polynomial base matrix was found by computational search for sparse lifted codes with nonzero rate and balanced X- and Z-distance behavior.For each lifting size, entries are reduced modulo D^L + 1 and used to form the CSS parity-check matrices.Source: §V · p. 4

Code family

• Changing the lifting size produces structured finite-length QLDPC codes with different block lengths, dimensions, and distance estimates.Table I summarizes the resulting code parameters.Source: §V · p. 4, Table I · p. 4

Decoder evaluation

• The simulations compare QBP, QBPGD, and RL-S using block error rate over the depolarizing channel.Block error rate counts decoder failures to return a consistent syndrome or a recovery in the wrong logical coset.Source: §V · p. 4

L = 17

• For the [[425, 21, 18]] code at L = 17, BP-based methods decode effectively over the depolarizing channel.Figure 1 compares QBP, QBPGD, and RL-S with different iteration budgets.Source: §V · p. 5, Figure 1 · p. 5

L = 24

• For the [[600, 32, d ≤20]] code at L = 24, RL-S improves block error rate relative to QBP, with its larger iteration budget performing best among plotted decoders.The code has dX, dZ, d ≤20 from a minimum-distance search.Source: §V · p. 5, Figure 2 · p. 5

L = 83

• For the [[2075, 87, d ≤34]] code at L = 83, QBP and QBPGD show a severe error floor, whereas RL-S does not.The graph contains 30,378 4-cycles, and the improvement is reported as consistent with prior RL decoding results for polar codes.Source: §V · p. 5, Figure 3 · p. 5

• Conjugate-compatible circulant lifting enables the CSS commutation condition HXH^T_Z = 0 to be proved directly in the polynomial domain.The proof occurs before binary expansion, simplifying verification of the resulting binary parity-check matrices.Source: §VI · p. 5
• The formulation connects classical convolutional-code polynomial descriptions with finite-length lifted-product QLDPC construction.The construction uses a constraint length 7 NASA convolutional code and a 3 × 4 polynomial base matrix to produce structured finite-length codes.Source: §VI · p. 5
• Selected constructed codes were evaluated over the depolarizing channel with BP-based decoders and were compatible with practical iterative decoding methods.Source: §VI · p. 5
• The work is a description and verification framework for new LP-QLDPC code design, not a new asymptotic lifted-product result.Future directions include distance-search constraints, polynomial-base-matrix optimization, and syndrome-extraction schedules adapted to the structure.Source: §VI · p. 5