Here is a list of LDPC codes.

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Code Description
Accumulate-repeat-accumulate (ARA) code A generalization of the RA code in which the outer repetition-code encoding step is augmented with an accumulator acting on a fraction of the incoming bits. In addition, the code may be punctured after the final accumulating step.
Affine-permutation-matrix LDPC (APM-LDPC) code LDPC code whose parity-check matrix can be put into the form of a block matrix consisting of permutation submatrices representing the affine permutation group or the zero submatrix. Given a cyclic group \(\mathbb{Z}_r\), the affine permutation group is \(\mathbb{Z}_r \rtimes \mathbb{Z}_r^{\times}\), where \(\mathbb{Z}_r^{\times}\) is the multiplicative group of integers modulo \(r\). Such codes are often constructed by lifting certain protographs into such block matrices [1].
Algebraic LDPC code LDPC code whose parity check matrix is constructed explicitly (i.e., non-randomly) from a particular graph [2,3] or an algebraic structure such as a combinatorial design [4–6], balanced incomplete block design [7], a partial geometry [8], a generalized polygon [9,10], or a Latin square [11–13]. The extra structure and/or symmetry [14] of these codes can often be used to gain a better understanding of their properties.
Array-based LDPC (AB-LDPC) code QC-LDPC code constructed deterministically from a disk array code known as a B-code. Its parity-check matrix admits a compact representation [15] and is related to RS codes.
Block LDPC (B-LDPC) code Member of a particular class of irregular QC-LDPC codes with efficient encoders.
Cycle LDPC code An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns.
Difference-set cyclic (DSC) code Cyclic LDPC code constructed deterministically from a difference set. Certain DSC codes satisfy more redundant constraints than Gallager codes and thus can outperform them [16].
Expander code LDPC code whose parity-check matrix is derived from the adjacency matrix of a bipartite expander graph [17] such as a Ramanujan graph or a Cayley graph of a projective special linear group over a finite field [18,19]. Expander codes admit efficient encoding and decoding algorithms and yield an explicit (i.e., non-random) asymptotically good LDPC code family.
Extended IRA (eIRA) code A generalization of the IRA code in which the outer LDGM code is replaced by a random sparse matrix containing no weight-two columns.
Finite-geometry LDPC (FG-LDPC) code LDPC code whose parity-check matrix is the incidence matrix of points and hyperplanes in either a Euclidean or a projective geometry. Such codes are called Euclidean-geometry LDPC (EG-LDPC) and projective-geometry LDPC (PG-LDPC), respectively. Such constructions have been generalized to incidence matrices of hyperplanes of different dimensions [20].
Gallager (GL) code The first LDPC code. The rows of the parity-check matrix of this regular code are divided into equal subsets, and columns in the first subset are randomly permuted to yield the corresponding rows in subsequent subsets.
Hsu-Anastasopoulos LDPC (HA-LDPC) code A regular LDPC code obtained from a concatenation of a certain random regular LDPC code and a certain random LDGM code. An \((l,r,g)\)-HA-LDPC code can be written using punctured LDPC and LDGM parts, and it is dual to the corresponding \((r,l,g)\)-MN-LDPC code [21]. Using rate-one LDGM codes eliminates high-weight codewords while admitting an amount of low-weight codewords that asymptotically vanishes, allowing code families to achieve the GV bound with high probability.
Irregular LDPC code An LDPC code whose parity-check matrix has a variable number of entries in each row or column.
Irregular repeat-accumulate (IRA) code A generalization of the RA code in which the outer 1-in-3 repetition encoding step is replaced by an LDGM code. A simple version is when different bits in the RA block are repeated a different number of times.
Lazebnik-Ustimenko (LU) code LDPC code whose Tanner graph comes from a particular family of \(q\)-regular graphs [22] of known girth and relatively large stopping sets.
Low-density parity-check (LDPC) code A binary linear code with a sparse parity-check matrix. Often a member of an infinite family of \([n,k,d]\) codes for which the numbers of nonzero entries in each row and in each column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\).
MacKay-Neal LDPC (MN-LDPC) code A code whose parity-check matrix is constructed non-deterministically via the MacKay-Neal prescription.
Margulis LDPC code Member of a class of LDPC codes deterministically constructed from explicit sparse regular expander graphs. The underlying Margulis-Gabber-Galil graph family provides explicit expanders [2,23], yielding deterministic sparse parity-check matrices. Related explicit LDPC constructions [24] utilize Ramanujan graphs [18,19].
Multi-edge LDPC code Irregular LDPC code whose construction generalizes those of the original examples of irregular LDPC as well as RA codes.
Pinwheel code A geometrically local binary LDPC code defined on planar graphs obtained from the pinwheel tiling [25]. Both bits and checks live on vertices of the graph. If \(L_N\) is the graph Laplacian at generation \(N\), the undepleted check matrix is \(\tilde H_N=(L_N-\mathbb{I})\bmod 2\), and the actual parity-check matrix \(H_N\) is obtained by removing an evenly spaced fraction of boundary checks.
Protograph LDPC code Binary version of a \(q\)-ary protograph LDPC code. Its parity-check matrix can be written as a block matrix whose blocks are either sums of permutation matrices or the zero matrix.
Quasi-cyclic LDPC (QC-LDPC) code LDPC code that can be put into quasi-cyclic form. Its parity check matrix can be put into the form of a block matrix consisting of either circulant permutation sub-matrices or the zero sub-matrix. Such codes are often constructed by lifting certain protographs into such block matrices [1]. Their simple structure makes them useful for several wireless communication standards.
Regular LDPC code An LDPC code whose parity-check matrix has a fixed number of ones in each row and each column. Such a code is called \((j,k)\)-regular if each column has weight \(j\) (variable-node degree) and each row has weight \(k\) (check-node degree). If the parity-check matrix has \(n\) columns and \(m\) rows, then regularity implies \(nj = mk\).
Repeat-accumulate (RA) code An LDPC code whose parity-check matrix has weight-two columns arranged in a step-like pattern for its last columns [26].
Repeat-accumulate-accumulate (RAA) code Generalization of the RA code in which two accumulators and permutations are used.
Spatially coupled LDPC (SC-LDPC) code An LDPC code whose parity-check matrix is constructed by “spatially” coupling several copies of a regular LDPC parity-check matrix in chain-like fashion (or, more generally, in grid-like fashion) to yield a band matrix. A finite-length chain is then capped by imposing either open boundary conditions (yielding non-tail-biting SC-LDPC codes) or periodic boundary conditions (yielding tail-biting SC-LDPC codes); sometimes extra terminating vertices are added to the ends of the chain. Matrices corresponding to translationally invariant chains are called time-invariant, and otherwise are called time-varying. These codes can be constructed, e.g., using the lifting procedure or using edge-cutting vectors [27]. Spatial coupling can also be applied to MN-LDPC and HA-LDPC protographs, yielding bounded-density SC-MN and SC-HA families [21].
Tanner-Sridhara-Fuja (TSF) code Array QC-LDPC code constructed from a cyclically shifted identity matrix; see [28; Exam. 21.6.5].
Tornado code Linear binary erasure code that is a precursor to fountain codes and is built from a multilayer cascade of sparse bipartite graphs. Its encoding and decoding operations involve only XOR gates [29,30][31; Sec. 30.2].
\([7,3,4]\) simplex code Second-smallest nontrivial member of the simplex-code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping. As a simplex code, it is equidistant: every nonzero codeword has Hamming weight \(4\).
\(q\)-ary LDPC code A \(q\)-ary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]_q\) codes for which the number of nonzero entries in each row and column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\).
\(q\)-ary protograph LDPC code A \(q\)-ary LDPC code whose parity-check matrix is constructed using the lifting procedure applied to the incidence matrix of a sparse graph called, in this context, a protograph. An ability to assign non-binary edge weight called edge scaling can also be used in code construction.

References

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