We introduce the pair-partition (PP) construction of binary Calderbank--Shor--Steane quantum low-density parity-check codes from circulant permutation matrices. A square array of pair partitions imposes linear paired-difference equations on the CPM exponents and thereby guarantees CSS orthogonality. Pairing graphs derived from this array allow the combinatorial design to be screened before exponent search. We further give and prove a complete algorithm for verifying lower bounds on the quantum minimum distance of a fixed CSS lift. The algorithm searches for zero-syndrome vectors outside the opposing stabilizer row space, uses only rigorously valid pruning rules, and finds no vector through a prescribed weight if and only if the corresponding distance exceeds that weight. For fixed column weight, row weight, and search limit, cyclic symmetry makes the combinatorial support-enumeration bound independent of the lift size, although matrix preprocessing and row-space tests can still depend on the lift size. Thus the construction stage and the distance-verification stage are both specified by directly checkable finite procedures.
We introduce the pair-partition (PP) construction of binary Calderbank--Shor--Steane quantum low-density parity-check codes from circulant permutation matrices. A square array of pair partitions imposes linear paired-difference equations on the CPM exponents and thereby guarantees CSS orthogonality. Pairing graphs derived from this array allow the combinatorial design to be screened before exponent search. We further give and prove a complete algorithm for verifying lower bounds on the quantum minimum distance of a fixed CSS lift. The algorithm searches for zero-syndrome vectors outside the opposing stabilizer row space, uses only rigorously valid pruning rules, and finds no vector through a prescribed weight if and only if the corresponding distance exceeds that weight. For fixed column weight, row weight, and search limit, cyclic symmetry makes the combinatorial support-enumeration bound independent of the lift size, although matrix preprocessing and row-space tests can still depend on the lift size. Thus the construction stage and the distance-verification stage are both specified by directly checkable finite procedures.
We study square-base Calderbank--Shor--Steane (CSS) hypergraph-product codes as a finite-length class for regular high-girth quantum low-density parity-check (LDPC) design. For base matrices of small column weight, we give checkable conditions for regularity, rank deficiency, and short-cycle exclusion, and we present explicit column-weight-three and column-weight-four examples with Tanner girth 6 and 8. We also analyze circulant permutation matrix (CPM) lifts of this class. Using the standard voltage-sum criterion, we identify orthogonality-forced Tanner 8-cycles and show that CPM lifting cannot raise the Tanner girth beyond 8 when these cycles are present. As a representative finite-length instance, a randomized CPM lift of the girth-8 base construction gives a [[28800,62,18≤d≤192]] girth-8 (3,6)-regular CSS-LDPC code. Complete enumeration excludes nontrivial logical supports of weight at most 16 on both sides, giving dX,dZ≥18; explicit weight-192 representatives give dX,dZ≤192. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in 2.993×108 independent trials at depolarizing probability p=0.1402; the Wilson 95\% upper confidence bound is 1.28×10−8.
We construct a rate-2/3 quantum low-density parity-check (LDPC) code from a (3,18)-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree P=101. The resulting code is a Calderbank--Shor--Steane (CSS) code with parameters [[34542,23032,18]]. Its distance is established by a computer-assisted proof: an explicit nontrivial logical operator gives the upper bound, and a complete symmetry-reduced enumeration excludes every nonzero vector of weight below 18 in both parity-check kernels. We also prove that every affine-in-t CPM lift of the same base satisfying CSS orthogonality has distance at most 18 for every prime lift degree P>19. The construction has row weight 18 and column weight 3, and the Tanner graphs of HX and HZ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in 108 trials at p=0.01, and a finite-length frame error rate (FER) sweep estimates the transition near p=0.029.
We construct a quantum low-density parity-check code family from a length-512 Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are (3,8)-regular with girth 8. The base code has parameters [[512,174,8]]. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor P=32, which has parameters [[16384,4142,18≤d≤32]], under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most 16 on both sides, giving dX,dZ≥18. Explicit weight-32 non-stabilizer logical representatives give the upper bounds dX,dZ≤32. A belief-propagation decoder with post-processing achieved frame error rate about 10−8 at p=0.085; an independently observed logical residual of weight 40 is consistent with the sharper structural bound.
We construct complete canonical logical bases for qubit CSS codes by assigning quaternary coefficients to binary circulant permutation matrix pair-partition (CPM-PP) checks, constructing and normalizing logical representatives, and expanding the result into binary matrices. When the two check systems have invertible submatrices on disjoint column sets, cofactor representatives are canonically paired by the inverse of a single pairing polynomial. The resulting pairs span the entire logical space. When the pairing polynomial is a cyclic-shift monomial with coefficient one, normalization preserves the binary weights of the representatives. We state the construction for general block dimensions and CPM size, work through a corresponding example, and report the parameters, check ranks, and basis weights of seven binary codes. Representative examples have parameters [[320,80,14]], [[448,112,18]], and [[2048,512,24]]. All three have maximum check weight 10 on both the X and Z sides. Their canonical logical representatives have binary weights 25, 27, and 27, respectively, on both the X and Z sides.
This paper develops a two-branch multiplicative-coset construction for regular Calderbank--Shor--Steane (CSS) quantum low-density parity-check base matrices. For a target column weight J and an even row weight L, the method reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions over a finite field. A normalized exhaustive search for these conditions produces base matrices for several (J,L) pairs, so the construction is not tied to a single degree distribution. The construction separates the finite-length design into two stages: the base matrix fixes the degree distribution and the first girth constraints, and a cyclic lift randomizes edge connections subject to exact algebraic checks. As a detailed example, we carry one (3,10)-regular base through the lift and decoding stages. For this example, the selected 64-fold lift gives a code whose same-type Tanner graphs have girth at least eight, and it also excludes a specified weight-16 nondegenerate logical-support orbit. Complete support enumeration excludes all nontrivial logical representatives of weight at most 16 on both sides. The resulting instance is a [[10240,4108,18≤d≤32]] CSS code. For decoding, we use joint log-domain belief propagation together with low-complexity deterministic post-processing rules for small residual syndromes, including repairs for residual patterns with two unsatisfied checks. The frame error rate (FER) measurements provide finite-length decoding data for this detailed example; at depolarizing probability p=0.058, the post-processing FER is 1.0×10−7.
This paper investigates certified upper bounds on the minimum distance of an explicit family of Calderbank-Shor-Steane quantum LDPC codes constructed from affine permutation matrices. All codes considered here have active Tanner graphs of girth eight. Rather than attempting to prove a general lower bound for the full code distance, we focus on constructing low-weight non-stabilizer logical representatives, which yield valid upper bounds once they are verified to lie in the opposite parity-check kernel and outside the stabilizer row space. We develop a unified framework for such witnesses arising from latent row relations, restricted-lift subspaces including block-compressed, selected-fiber, and CRT-stripe constructions, cycle- 8 elementary trapping-set structures, and decoder-failure residuals. In every case, search is used only to generate candidates; the reported bounds begin only after explicit kernel and row-space exclusion tests have been passed. For the latent part, we also identify a block-compression criterion under which the certification becomes exact. Applying these methods to representative APM-LDPC codes sharpens previously reported upper bounds and provides concrete certified values across the explored parameter range.