Multi-Edge Type QC-LDPC Graph

• Multi-Edge Type QC-LDPC Graph is a structured multigraph formalism allowing multiple edge classes for flexible QC-LDPC code design.
• It employs circulant permutation and exponent matrices to model and control cycle structures, directly impacting girth and error performance.
• Construction methods integrate combinatorial designs, progressive edge-growth, and MET ensemble analysis to enhance decoding and low-SNR performance.

A multi-edge type QC-LDPC graph is the Tanner graph associated with a quasi-cyclic low-density parity-check code whose base description permits either multiple parallel edges between a check node and a variable node, or multiple edge classes whose incidences are fixed before lifting. In the quasi-cyclic realization, the parity-check matrix is assembled from circulant permutation matrices (CPMs) and zero blocks; in the multiple-edge case, a single block position may contain a sum of several CPMs, so the lifted object is naturally a structured bipartite multigraph rather than an ordinary simple graph (Park et al., 2012). In closely related multi-edge-type (MET) ensemble theory, the same phrase can instead denote a Tanner graph in which “Each node is characterized by the number of connections (sockets) to edges of each edge-type,” and that usage is broader than explicit QC lifting; one paper on MET-LDPC design for CV-QKD states directly that no explicit QC structure is used (Mani et al., 2018). This suggests that the subject sits at the intersection of protograph lifting, graph-cycle control, ensemble threshold analysis, and hardware-oriented structured decoding.

1. Terminological scope and defining models

In the protograph-based QC-LDPC literature, the starting point is a block parity-check matrix

$$H=\left[\begin{array}{cccc} H_{00}&H_{01}&\cdots &H_{0(n-1)}\ H_{10}&H_{11}&\cdots &H_{1(n-1)}\ \vdots &\vdots &\ddots &\vdots \ H_{(m-1)0}&H_{(m-1)1}&\cdots &H_{(m-1)(n-1)} \end{array}\right],$$

where each block is either zero or circulant. A protograph is encoded by the incidence matrix

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$

so $p_{j,l}\ge 2$ means that the protograph contains multiple edges between check node $j$ and variable node $l$ (Park et al., 2012).

A more explicit CPM-based formulation writes each nonzero block as

$$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$

with distinct shifts $p^r_{ij}\in\{0,1,\dots,N-1\}$. If $L=\max_{i,j}\{l\}$, the code is called an $m$-level Type-$L$ CPM-QC-LDPC code; $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$0 yields a single-edge protograph code, while $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$1 yields a multiple-edge protograph code (Tasdighi et al., 2017).

MET ensemble theory uses a different but overlapping language. There, the graph is specified by node classes and edge types rather than only by multiplicities in a base matrix. With $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$2 edge types, a variable or check node is described by a degree vector $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$3, and the ensemble is encoded by multivariable degree polynomials such as

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$4

In this vocabulary, a protograph is a special case in which no two edges incident to the same VN or CN share the same type (Mani et al., 2018, Paolini et al., 2011).

2. Matrix, exponent, and edge-type representations

The algebraic representation of a multi-edge type QC-LDPC graph is usually mediated by an exponent matrix. In the CPM-QC setting, the parity-check matrix is uniquely represented by

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$5

where

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$6

for a nonzero block and $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$7 for a zero block (Tasdighi et al., 2017).

An equivalent base-matrix/exponent-matrix description uses an integer base matrix $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$8 and a set-valued exponent matrix $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$9. If $p_{j,l}\ge 2$0, then

$p_{j,l}\ge 2$1

with distinct entries; if $p_{j,l}\ge 2$2, then $p_{j,l}\ge 2$3. The lifted QC block is then

$p_{j,l}\ge 2$4

This formulation makes the multigraph interpretation explicit: a single base entry may correspond to several parallel circulant edges (Amirzade et al., 2021).

For ordinary QC-LDPC codes, the exponent matrix may also be scalar-valued: $p_{j,l}\ge 2$5 with $p_{j,l}\ge 2$6, where $p_{j,l}\ge 2$7 denotes a zero block and nonnegative entries denote CPM shifts. The corresponding protograph mother matrix $p_{j,l}\ge 2$8 is obtained by replacing $p_{j,l}\ge 2$9 by $j$0 and other entries by $j$1 (Usatyuk et al., 2024).

In MET ensemble analysis, the same graph is described in edge perspective by normalized derivatives such as

$j$2

subject to the edge-consistency constraints

$j$3

The graph can therefore be read either as a lifted block-circulant multigraph or as a socket-typed ensemble, depending on the design layer under study (Mani et al., 2018).

3. Cycle structure, girth, and inevitable subgraphs

The central structural problem for multi-edge type QC-LDPC graphs is the control of short cycles. In exponent form, the standard multiple-edge QC-LDPC cycle condition is

$j$4

with $j$5, where $j$6 select specific CPMs inside the multi-edge blocks (Tasdighi et al., 2017). An equivalent formulation used elsewhere is

$j$7

which is the basis for difference-matrix cycle screening (Amirzade et al., 2021).

Some cycles are unavoidable regardless of lift size and shift assignment. In the graph-theoretic treatment of multiple-edge protographs, these are induced by specific irreducible inevitable-cycle-inducing subgraphs. The complete classification is given by theta graphs $j$8 and dumbbell graphs $j$9, whose simple abelian-forcing tailless non-reversing closed walk lengths are

$l$0

Avoiding $l$1 is necessary for girth at least $l$2, and avoiding the $l$3 patterns as well is needed for girth at least $l$4 (Park et al., 2012).

A separate algebraic route analyzes girth through the matrix sequence

$l$5

and the associated square matrix $l$6 extracted from $l$7. For protograph-based QC-LDPC matrices, the criterion

$l$8

yields necessary and sufficient conditions for girths between $l$9 and $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$0; for girth larger than $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$1, multi-step lifting or pre-lifting is used to break the standard circulant limitation (Smarandache et al., 2021).

Difference matrices $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$2 and $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$3 provide a third framework. In this setting, $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$4 records pairwise differences inside exponent tuples, and $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$5 records differences of those differences. These matrices yield necessary and sufficient conditions for single-edge QC-LDPC codes with girth $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$6, $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$7, and $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$8, and they also produce, for the first time in that work, a lower bound on the lifting degree for regular and irregular multiple-edge QC-LDPC codes with girth $$H_{ij}=I^{p^{1}_{ij}}+I^{p^{2}_{ij}}+\cdots+I^{p^{l}_{ij}},$$9. The same approach reduces the conditions for considering $p^r_{ij}\in\{0,1,\dots,N-1\}$0-cycles in the multiple-edge setting from seven states to five states (Amirzade et al., 2017).

4. Construction methodologies

One major design line uses combinatorial designs to generate the base multigraph. Perfect difference families (PDFs) and quasi-perfect difference families (QPDFs) produce $p^r_{ij}\in\{0,1,\dots,N-1\}$1-level Type-$p^r_{ij}\in\{0,1,\dots,N-1\}$2 CPM-QC-LDPC codes via Construction 1. If the underlying design is a PDF and $p^r_{ij}\in\{0,1,\dots,N-1\}$3, or a QPDF with $p^r_{ij}\in\{0,1,\dots,N-1\}$4, the resulting code has girth at least $p^r_{ij}\in\{0,1,\dots,N-1\}$5. Construction 2 then applies the column dispersion technique (CDT), which redistributes the same CPM shifts vertically across several levels; if $p^r_{ij}\in\{0,1,\dots,N-1\}$6 is obtained from $p^r_{ij}\in\{0,1,\dots,N-1\}$7 by $p^r_{ij}\in\{0,1,\dots,N-1\}$8-CDT, then the new code satisfies

$p^r_{ij}\in\{0,1,\dots,N-1\}$9

A notable empirical conclusion is that for short to moderate lengths, “the higher the number of short simple cycles is, the better (sharper) the waterfall is,” although too many short cycles can worsen the error floor (Tasdighi et al., 2017).

A second construction line uses trades from super-simple directed designs. The trade-based method forms a sparse matrix $L=\max_{i,j}\{l\}$0 from pair–block incidences, then uses row-shifted copies to build a base matrix $L=\max_{i,j}\{l\}$1 for a multiple-edge protograph. The key structural condition is the absence of any all-one $L=\max_{i,j}\{l\}$2 submatrix in the concatenated seed matrix $L=\max_{i,j}\{l\}$3. A theorem identifies cyclical trades $L=\max_{i,j}\{l\}$4 with Tanner-graph $L=\max_{i,j}\{l\}$5-cycles, so the smallest cyclical trade volume controls girth directly. The same structure removes the $L=\max_{i,j}\{l\}$6 term from the girth-$L=\max_{i,j}\{l\}$7 lifting-degree bound and the $L=\max_{i,j}\{l\}$8 term from the girth-$L=\max_{i,j}\{l\}$9 bound, and it reduces exponent-matrix search dramatically; one example in the paper states that a generic $m$0-regular QC-LDPC design may require search space $m$1, whereas the trade-based method reduces it to $m$2 (Amirzade et al., 2021).

A third line uses explicit avoidance of inevitable-cycle-inducing subgraphs. For regular multiple-edge protographs of girth at least $m$3 or $m$4, balanced ternary designs, Steiner systems, configurations, difference triangle sets, and pairwise balanced designs are used to construct incidence matrices free of the required $m$5 patterns. The same work adapts a greedy shift-assignment algorithm to the multi-edge setting by choosing column shifts as

$m$6

with $m$7 from $m$8, and then selecting the smallest lift $m$9 that keeps all relevant tailless non-reversing closed walks of lengths $L$0 at nonzero shift sum modulo $L$1 (Park et al., 2012).

Progressive edge-growth methods provide a more local, graph-construction view. The metric-constrained PEG framework unifies PEG and ACE-constrained PEG through

$L$2

and then extends to multi-edge lookahead through the multi-edge local girth and the edge-trials parameter $L$3. For a candidate check node $L$4, the $L$5-edge local girth

$L$6

evaluates not just the current edge but the best achievable local girth after several future insertions. In the QC version, one inserts an entire cyclic orbit

$L$7

rather than a single Tanner edge, and the algorithm detects cycles that QC-PEG and CP-PEGA can miss. The reported QC-specific examples include undetectable $L$8-cycles in QC-PEG and undetectable $L$9-cycles in CP-PEG; increasing edge-trials improves cycle structure and often BER (He et al., 2016).

5. Ensemble analysis, stability, and threshold optimization

When the same graph family is viewed as a MET ensemble, asymptotic analysis shifts from girth conditions to density evolution and EXIT-type tools. In the CV-QKD setting, MET-LDPC codes are designed through multilevel coding and multistage decoding, with levelwise rates constrained by

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$00

and convergence analyzed by generalized EXIT charts. The MET density-evolution recursion is

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$01

while the G-EXIT and dual G-EXIT curves are used as a fast threshold test. Example designed rates $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$02, $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$03, and $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$04 were reported within $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$05 dB, $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$06 dB, and $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$07 dB of capacity, respectively; the same paper emphasizes that this is MET-LDPC Tanner-graph design rather than an explicit QC construction (Mani et al., 2018).

A local dynamical-systems view is given by the stability theory of MET D-GLDPC ensembles on the BEC. If $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$08 captures the variable-node weight-2 contribution and $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$09 the check-node weight-2 contribution, then the erasure-free fixed point is locally stable if and only if

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$10

where $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$11 is the spectral radius. In the single-edge case this reduces to the classical scalar condition $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$12, while protograph-like representations force $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$13 and $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$14 because no two sockets at the same node share the same edge type (Paolini et al., 2011).

Threshold optimization of the edge-type structure can then be posed as a constrained search over degree distributions or node-type fractions. One reported approach is Adaptive Range (AR), which restricts the search around the current best candidate and adapts the local range according to the gap between the best and next-best solutions. The method is applied to both irregular LDPC and MET-LDPC ensembles; for MET-LDPC, the recommendation is AR for inner degree-distribution optimization and Differential Evolution for outer structure search. A cited example reports an improvement from $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$15 to $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$16 on the BI-AWGN channel (Jayasooriya et al., 2016).

6. Decoding architectures, performance trade-offs, and application domains

The quasi-cyclic form of the graph is particularly important for implementation. In a GPU layered decoder for QC-METLDPC codes in CV-QKD, the base matrix entries are either a non-negative integer, denoting a $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$17 circulant permutation matrix obtained by cyclically shifting the identity matrix, or $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$18, denoting a zero block. Layered belief propagation updates

$P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$19

at initialization, then reuses updated messages immediately from one layer to the next. The implementation stores a compact parity-check representation, merges unrelated submatrices, and decodes multiple codewords simultaneously on an NVIDIA TITAN Xp GPU. For block length $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$20, expansion factor $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$21, and $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$22 simultaneously decoded codewords without early termination, the reported throughputs are $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$23 Mbits/s at rate $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$24, $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$25 Mbits/s at rate $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$26, and $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$27 Mbits/s at rate $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$28 (Li et al., 2020).

The principal application domain in the cited MET/QC-MET literature is low-SNR reconciliation for continuous-variable quantum key distribution. MET-LDPC codes are described there as particularly suitable for very low code rates because of degree-one variable nodes, and the practical design target is the low-SNR regime needed for long-distance CV-QKD. In that context, QC regularity supports parallel hardware decoding, while MET flexibility supports very low thresholds (Mani et al., 2018, Li et al., 2020).

Performance comparisons across the literature show that graph structure is not governed by a single scalar criterion. Larger edge-trials in MM-PEGA and MM-QC-PEGA improve ACE spectra, VN-local-girth distributions, and BER in the reported simulations (He et al., 2016). Multiple-edge protographs can yield larger upper bounds on minimum Hamming distance than comparable single-edge protographs, while structured QC constructions can perform as well as PEG LDPC and PEG QC-LDPC comparators under BP decoding over the BIAWGN channel (Park et al., 2012). At the same time, the combinatorial-design literature reports a nuanced finite-length trade-off: higher multiplicities of short simple cycles can sharpen the waterfall, but excessive short cycles can worsen the error floor (Tasdighi et al., 2017).

A broader extension appears in work that explicitly generalizes a QC-LDPC Tanner graph to a multigraph by allowing block entries such as $P=[p_{j,l}], \qquad p_{j,l}=\mathrm{wt}(H_{j,l}),$29, then reuses that representation in spatially coupled, toric, and hyperbolic settings. That usage remains faithful to the same block-circulant graph model—CPMs, exponent matrices, base graphs, and trapping sets—but exports it beyond conventional coding problems (Usatyuk et al., 2024). This suggests that the multi-edge type QC-LDPC graph is best understood not merely as a parity-check pattern, but as a structured multigraph formalism in which local edge multiplicity, global cyclic symmetry, and edge-class semantics can be tuned independently.