Here is a list of projective codes.

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Code Description
Cycle LDPC code An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns.
Cycle code A code whose parity-check matrix is obtained from the incidence matrix of a graph over \(\mathbb{F}_2\). This code’s properties are derived from the size two chain complex associated with the graph. Not every binary linear code is homological, but there exist homological families that asymptotically saturate the Hamming bound [1].
Denniston code Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{2^a}\) codes for \(i < a\) constructed using Denniston arcs [2; Sec. 19.7.3].
Hill projective-cap code Member of a projective code family that contains two \(q\)-ary sharp configurations [3; Table 12.1] and that is constructed using projective caps.
Hirschfeld code A \([q+1,4,q-2]_q\) projective geometry code for non-prime \(q\) that is an example of an MDS code that is not an RS code; see [4; Exam. 7.6] for the generator matrix.
Hoffman-Singleton cycle code A \([50,21,12]\) cycle code whose parity-check matrix is the incidence matrix of the Hoffman-Singleton graph [5]. Its dual is a \([50,29,8]\) code [6; Table II].
Hyperoval code Projective code constructed from a hyperoval in the projective plane \(PG(2,q)\), where \(q\) is even. Since a hyperoval is a set of \(q+2\) points with no three collinear, the corresponding projective code has parameters \([q+2,3,q]_q\) [2; Exam. 19.2.1]; the \([6,3,4]_4\) hexacode is the smallest example.
Incidence-matrix projective code A projective code whose generator matrix is the incidence matrix of points and hyperplanes in a projective space. Has been generalized to incidence matrices of other structures [7,8][9; Sec. 14.4]. More generally, columns of a code’s parity-check matrix can also be organized as an incidence matrix.
Laplacian code A binary linear code whose parity-check matrix is the graph Laplacian reduced mod 2. For an undirected graph \(\Gamma\) with degree matrix \(D\) and adjacency matrix \(A\), the parity-check matrix is the symmetric matrix \(H=(D-A)\bmod 2\).
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 [10,11], yielding deterministic sparse parity-check matrices. Related explicit LDPC constructions [12] utilize Ramanujan graphs [13,14].
Ovoid code Member of a \([q^2+1,4,q^2-q]_q\) projective two-weight code family obtained from ovoids in \(\mathrm{PG}(3,q)\). If the columns of a generator matrix are the \(q^2+1\) points of an ovoid, then every hyperplane meets the ovoid in either \(1\) or \(q+1\) points, yielding the two nonzero weights \(q^2\) and \(q^2-q\). See [15; pg. 107][16; pg. 192] for further details.
Projective geometry code Linear \(q\)-ary \([n,k,d]\) code such that columns of its generator matrix \(G\) do not contain any repeated columns or the zero column. That way, each column corresponds to a distinct point in the projective space \(PG(k-1,q)\) arising from a \(k\)-dimensional vector space over \(\mathbb{F}_q\). A choice of \(k\) linearly independent columns determines an information set. Columns of a code’s parity-check matrix can similarly correspond to points in projective space. This formulation yields connections to projective geometry, which can be applied to determine code properties.
Projective two-weight code A projective code whose codewords all have one of two possible nonzero Hamming weights.
\([10,5,6]_9\) Glynn code The unique trace-Hermitian self-dual \([10,5,6]_9\) code, constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve.
\([15,6,5]\) Petersen cycle code A \([15,6,5]\) cycle code whose parity-check matrix is the incidence matrix of the Petersen graph. The Petersen graph can be thought of as a dodecahedron with antipodes identified [17; Appx. A.2.1].
\([2^m-1,m,2^{m-1}]\) simplex code A member of the equidistant code family that is dual to the \([2^m-1,2^m-m-1,3]\) Hamming family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,2)\), with each column being a chosen representative of the corresponding element. Simplex codes saturate the Plotkin bound and hence have nonzero codewords all of the same weight, \(2^{m-1}\) [18; Th. 11(a)][19; Thm. 1.10.5]. The codewords form a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping.
\([4,2,3]_3\) Tetracode The \([4,2,3]_3\) ternary self-dual MDS code that has connections to lattices [20]. Its weight enumerator is the Gleason polynomial \(g_4\) [21; Rem. 4.2.6].
\([56,6,36]_3\) Hill-cap code Projective two-weight ternary code based on the Games graph [22][2; Table 19.1] and Hill’s 56-cap [23]. Its automorphism group contains \(PSL_3(4)\) [24].
\([6,3,4]_4\) Hexacode The \([6,3,4]_4\) Hermitian self-dual MDS code that has connections to projective geometry, lattices [20], and conformal field theory [25]. Its weight enumerator is the Gleason polynomial \(g_7\) [21; Rem. 4.2.6].
\([7,3,4]\) 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 \(4\).
\([78,6,56]_4\) Hill-cap code Projective two-weight quaternary code based on a cap that corresponds to a strongly regular graph [22; Table 7.1].
\(q\)-ary simplex code An \([n,m,q^{m-1}]_q\) equidistant projective code with \(n=\frac{q^m-1}{q-1}\), denoted as \(S(q,m)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,q)\), with each column being a chosen representative of the corresponding element. All nonzero simplex codewords have a constant weight of \(q^{m-1}\) [26,27].

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