Here is a list of codes related to combinatorial designs.
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].
Dodecacode There exists a \(5\)-\((12, 6, 3)\) design in the dodecacode, and a \(3\)-\((11, 5, 4)\) design in the shortened dodecacode [7].
Dual linear code Linear codes and their duals are related to combinatorial designs via the Assmus-Mattson theorem [8,9] (see [10; 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].
Golay code The supports of the weight-seven (weight-eight) codewords of the (extended) Golay code support the Steiner system \(S(4,7,23)\) (\(S(5,6,12)\)) [15,16][17; pg. 89]. Its blocks are called octads.
Hadamard code Hadamard designs are combinatorial designs constructed from Hadamard matrices [18]; see Ref. [19].
Higman-Sims graph-adjacency code Codewords of weight 36 of the Higman-Sims graph-adjacency code form a \(2\)-\((100,36,525)\) design [20; 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 [20; Prop. 1.1].
Jump code Certain types of combinatorial designs can be used to obtain jump codes [21–23].
Lexicographic code Some lexicodes yield Steiner systems [24].
Mixed code Combinatorial designs have been generalized to mixed alphabets [25].
Nordstrom-Robinson (NR) code NR codewords give \(3\)-\((16, 6, 4)\), \(3\)-\((16, 8, 3)\), and \(3\)-\((16, 10, 24)\) designs [26; pg. 164].
Perfect code Perfect codes and combinatorial designs are related [27,28].
Perfect-tensor code Combinatorial designs and \(d\)-uniform quantum states are related [29].
Pless symmetry code The supports of fixed-weight codewords of certain Pless symmetry codes support combinatorial designs [16,30,31].
Preparata code Preparata codewords of each weight form a 3-design [26; pg. 471].
Reed-Muller (RM) code Fixed-weight RM codewords of weight less than \(2^m\) support combinatorial 3-designs [10; Exam. 5.2.7].
Self-dual linear code Self-dual extremal codes yield combinatorial \(\leq 5\)-designs using the Assmus-Mattson theorem [8] (see [10; Sec. 5.4]). See [32; 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 \(S(4,5,11)\) (\(S(5,6,12)\)) [15,16][17; pg. 89]. Its blocks are called hexads.
\([2^r,2^r-r-1,4]\) Extended Hamming code Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [17; 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 \(S(2,3,2^r-1)\) [17; 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 [8]. There are several designs associated with this code [33].
\([7,4,3]\) 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 [10; 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.
\(q\)-ary quadratic-residue (QR) code The supports of fixed-weight codewords of certain \(q\)-ary QR codes support combinatorial designs [8,16,27].

References

[1]
S. J. Johnson and S. R. Weller, “Regular low-density parity-check codes from combinatorial designs”, Proceedings 2001 IEEE Information Theory Workshop (Cat. No.01EX494) DOI
[2]
S. J. Johnson and S. R. Weller, “Construction of low-density parity-check codes from Kirkman triple systems”, GLOBECOM’01. IEEE Global Telecommunications Conference (Cat. No.01CH37270) DOI
[3]
S. J. Johnson and S. R. Weller, “Resolvable 2-designs for regular low-density parity-check codes”, IEEE Transactions on Communications 51, 1413 (2003) DOI
[4]
C. Ding and C. Tang, “Infinite families of near MDS codes holding \(t\)-designs”, (2019) arXiv:1910.08265
[5]
C. Tang and C. Ding, “An infinite family of linear codes supporting 4-designs”, (2020) arXiv:2001.00158
[6]
C. Ding, C. Tang, and V. D. Tonchev, “The Projective General Linear Group \(\mathrm{PGL}_2(\mathrm{GF}(2^m))\) and Linear Codes of Length \(2^m+1\)”, (2020) arXiv:2010.09448
[7]
J. Kim and V. Pless, Designs, Codes and Cryptography 30, 187 (2003) DOI
[8]
E. F. Assmus Jr. and H. F. Mattson Jr., “New 5-designs”, Journal of Combinatorial Theory 6, 122 (1969) DOI
[9]
A. R. Calderbank, P. IDelsarte, and N. J. A. Sloane, “A strengthening of the Assmus-Mattson theorem”, IEEE Transactions on Information Theory 37, 1261 (1991) DOI
[10]
V. D. Tonchev, "Codes and designs." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[11]
Y. Fujiwara, D. Clark, P. Vandendriessche, M. De Boeck, and V. D. Tonchev, “Entanglement-assisted quantum low-density parity-check codes”, Physical Review A 82, (2010) arXiv:1008.4747 DOI
[12]
P. A. H. Bours, “On the construction of perfect deletion-correcting codes using design theory”, Designs, Codes and Cryptography 6, 5 (1995) DOI
[13]
A. Mahmoodi, Designs, Codes and Cryptography 14, 81 (1998) DOI
[14]
D. J. C. MacKay and M. C. Davey, “Evaluation of Gallager Codes for Short Block Length and High Rate Applications”, Codes, Systems, and Graphical Models 113 (2001) DOI
[15]
L. J. Paige, “A Note on the Mathieu Groups”, Canadian Journal of Mathematics 9, 15 (1957) DOI
[16]
M. HUBER, “CODING THEORY AND ALGEBRAIC COMBINATORICS”, Selected Topics in Information and Coding Theory 121 (2010) arXiv:0811.1254 DOI
[17]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[18]
J. A. Todd, “A Combinatorial Problem”, Journal of Mathematics and Physics 12, 321 (1933) DOI
[19]
C. J. Colbourn and J. H. Dinitz, editors , Handbook of Combinatorial Designs (Chapman and Hall/CRC, 2006) DOI
[20]
V. D. Tonchev, “Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs”, IEEE Transactions on Information Theory 43, 1021 (1997) DOI
[21]
G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation”, Physical Review A 68, (2003) arXiv:quant-ph/0208140 DOI
[22]
T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, Designs, Codes and Cryptography 29, 51 (2003) DOI
[23]
Y. Lin and M. Jimbo, “Extremal properties of t-SEEDs and recursive constructions”, Designs, Codes and Cryptography 73, 805 (2013) DOI
[24]
J. Conway and N. Sloane, “Lexicographic codes: Error-correcting codes from game theory”, IEEE Transactions on Information Theory 32, 337 (1986) DOI
[25]
W. J. Martin, Designs, Codes and Cryptography 16, 271 (1999) DOI
[26]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[27]
E. F. Assmus, Jr. and H. F. Mattson, Jr., “Coding and Combinatorics”, SIAM Review 16, 349 (1974) DOI
[28]
K. T. Phelps, “Combinatorial designs and perfect codes”, Electronic Notes in Discrete Mathematics 10, 220 (2001) DOI
[29]
D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski, “Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices”, Physical Review A 92, (2015) arXiv:1506.08857 DOI
[30]
V. Pless, “The Weight of the Symmetry Code for p=29 and the 5‐Designs Contained Therein”, Annals of the New York Academy of Sciences 175, 310 (1970) DOI
[31]
V. Pless, “Symmetry codes over GF(3) and new five-designs”, Journal of Combinatorial Theory, Series A 12, 119 (1972) DOI
[32]
C. J. Colbourn, editor , CRC Handbook of Combinatorial Designs (CRC Press, 2010) DOI
[33]
M. Harada, A. Munemasa, and V. D. Tonchev, “A Characterization of Designs Related to an Extremal Doubly-Even Self-Dual Code of Length 48”, Annals of Combinatorics 9, 189 (2005) DOI