Combinatorial design code 


A constant-weight binary code that is mapped into a combinatorial \(t\)-design.

The mapping proceeds as follows for a length-\(n\) code with codewords of constant weight \(w\). A codeword \(c\) corresponds to a block of the design, with the codeword's \(j\)th coordinate labeling whether or not element \(j\) is contained in the block. There are a total of \(n\) elements, and the constant weight of the code implies that each block contains a fixed number \(w\) of elements. The code supports an \(S(t,w,n)\) Steiner system if each subset of \(t\leq w\) elements is contained in exactly one block. More generally, the code supports a combinatorial \(t\)-design, or, more precisely, a \(t\)-\((n,w,\lambda)\)-design, if each such \(t\)-subset is contained in exactly \(\lambda \geq 1\) blocks.

For example, the eight codewords of weight \(w=3\) of the \([7,4,3]\) Hamming code support a \(2\)-\((7,3,1)\)-design a.k.a. an \(S(2,3,7)\) Steiner system. The codeword \(1000110\) corresponds to a block containing elements 1, 5, and 6. Similarly, the other seven codewords correspond to blocks 257, 367, 147, 246, 345, and 123. Each pair of elements is contained in exactly one block.

Combinatorial \(t\)-designs exist for all \(t\) [1].


See [2] for a review combinatorial designs.See [35] for books on combinatorial designs.




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T. Beth, D. Jungnickel, and H. Lenz, Design Theory (Cambridge University Press, 1999) DOI
C. J. Colbourn and J. H. Dinitz, editors , Handbook of Combinatorial Designs (Chapman and Hall/CRC, 2006) DOI
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C. J. Colbourn, editor , CRC Handbook of Combinatorial Designs (CRC Press, 2010) DOI
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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
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
V. D. Tonchev, “Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs”, IEEE Transactions on Information Theory 43, 1021 (1997) DOI
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
W. J. Martin, Designs, Codes and Cryptography 16, 271 (1999) DOI
J. Kim and V. Pless, Designs, Codes and Cryptography 30, 187 (2003) DOI
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Zoo Code ID: combinatorial_design

Cite as:
“Combinatorial design code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_combinatorial_design, title={Combinatorial design code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Combinatorial design code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.