Here is a list of codes related to combinatorial designs.

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Code Relation
Algebraic LDPC code Combinatorial designs can be used to construct explicit LDPC codes [1–3].
Binary BCH code A family of BCH codes supports an infinite family of combinatorial 4-designs [4,5].
Combinatorial design
Cyclic linear \(q\)-ary code Two families of cyclic \(q\)-ary codes support an infinite family of combinatorial 3-designs [6]. The supports of all fixed-weight codewords of a \(q\)-ary cyclic code support a combinatorial \(1\)-design [7; Corr. 5.2.4].
Dodecacode There exists a \(5\)-\((12, 6, 3)\) design in the dodecacode, and a \(3\)-\((11, 5, 4)\) design in the shortened dodecacode [8].
Dual linear code Linear codes and their duals are related to combinatorial designs via the Assmus-Mattson theorem [9,10] (see [7; Sec. 5.4]).
EA combinatorial-design QLDPC code Combinatorial designs can be used to construct EA QLDPC codes [11].
Editing code Perfect deletion correcting codes can be constructed using combinatorial design theory [12,13].
Gallager (GL) code Some Steiner systems can be used to construct Gallager codes [14].
Hadamard code Hadamard designs are combinatorial designs constructed from Hadamard matrices [15]; see Ref. [16].
Higman-Sims graph-adjacency code Codewords of weight 36 of the Higman-Sims graph-adjacency code form a \(2\)-\((100,36,525)\) design [17; Remark 1.7]
Hoffman-Singleton cycle code The incidence matrix of the Hoffman-Singleton graph can be converted into a \(2\)-\((50,14,13)\) design [17; Prop. 1.1].
Jump code Certain types of combinatorial designs can be used to obtain jump codes [18–20].
Lexicographic code Some lexicodes yield Steiner systems [21].
Mixed code Combinatorial designs have been generalized to mixed alphabets [22].
Nordstrom-Robinson (NR) code NR codewords give \(3\)-\((16, 6, 4)\), \(3\)-\((16, 8, 3)\), and \(3\)-\((16, 10, 24)\) designs [23; pg. 164].
Perfect code Perfect codes and combinatorial designs are related [24,25].
Perfect-tensor code Combinatorial designs and \(d\)-uniform quantum states are related [26–28].
Pless symmetry code The supports of fixed-weight codewords of certain Pless symmetry codes support combinatorial designs [29–31].
Preparata code Preparata codewords of each weight form a 3-design [23; pg. 471].
Pseudo Golay code Supports of codewords of any fixed symmetrized type of pseudo Golay codes form a 5-design [32–34].
Quadratic-residue (QR) code The supports of fixed-weight codewords of certain \(q\)-ary QR codes support combinatorial designs [9,24,31].
Quaternary Golay code Supports of codewords of any fixed symmetrized type of the quaternary Golay code form a 5-design [32–34].
Reed-Muller (RM) code Fixed-weight RM codewords of weight less than \(2^m\) support combinatorial 3-designs [7; Exam. 5.2.7].
Self-dual linear code Self-dual extremal codes yield combinatorial \(\leq 5\)-designs using the Assmus-Mattson theorem [9] (see [7; Sec. 5.4]). See [35; Table 1.61, pg. 683] for a table of combinatorial designs obtained from self-dual codes.
Ternary Golay code The supports of the weight-five (weight-six) codewords of the (extended) ternary Golay code support the Steiner system \(S(4,5,11)\) (\(S(5,6,12)\)) [31,36][37; pg. 89]. Its blocks are called hexads.
\([23, 12, 7]\) Golay code The supports of the weight-seven codewords of the Golay code support the Steiner system \(S(4,7,23)\) [31,36][37; pg. 89]. Its blocks are called octads.
\([24, 12, 8]\) Extended Golay code The supports of the weight-eight codewords of the extended Golay code support the Steiner system \(S(5,6,12)\) [31,36][37; pg. 89]. Its blocks are called octads.
\([2^r,2^r-r-1,4]\) Extended Hamming code Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [37; pg. 89].
\([2^r-1,2^r-r-1,3]\) Hamming code Weight-three codewords of the \([2^r-1,2^r-r-1, 3]\) Hamming code support the Steiner system \(S(2,3,2^r-1)\) [37; pg. 89].
\([48,24,12]\) self-dual code Fixed-weight codewords of extremal self-dual doubly even codes whose length divides 24 form a combinatorial 5-design [9]. There are several designs associated with this code [38].
\([7,4,3]\) Hamming code Weight-three and weight-four codewords of the \([7,4,3]\) Hamming code support combinatorial \(2\)-\((7,3,1)\) and \(2\)-\((7,4,2)\) designs, respectively [7; Exam. 5.2.5].
\(q\)-ary code Designs can be constructed from \(q\)-ary codes by taking the supports of a subset of codewords of constant weight.

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