## Description

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. A combinatorial \(2\)-design with two block intersection sizes is called a quasi-symmetric design. A \(3\)-\((q^d+1,q+1,1)\) combinatorial design is sometimes called a Witt design (a.k.a. a spherical geometry design) [1][2; Remark 6.10].

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 \(t\) [3,4]. Existence of certain quasi-symmetric designs has also been established [5].

## Notes

## Parents

- Binary code — If the number of a code is less than or equal to its dual distance, then some sets of fixed-weight codewords form a combinatorial design [11; Thm. 6.7].
- Constant-weight code
- Subspace design — Combinatorial designs are designs on a space of fixed-weight binary strings (a.k.a. Johnson association scheme) [12,13]. Subspace designs reduce to combinatorial designs for \(q=2\).

## Cousins

- Galois-field \(q\)-ary code — Designs can be constructed from \(q\)-ary codes by taking the supports of a subset of codewords of constant weight.
- Cyclic linear \(q\)-ary code — The supports of all fixed-weight codewords of a \(q\)-ary cyclic code support a combinatorial \(1\)-design [7; Corr. 5.2.4].
- Reed-Muller (RM) code — Fixed-weight RM codewords of weight less than \(2^m\) support combinatorial 3-designs [7; Ex. 5.2.7].
- \([7,4,3]\) 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; Ex. 5.2.5].
- \([2^r-1,2^r-r-1,3]\) Hamming code — Weight-three codewords of the \([2^r-1,2^r-r-1, 3]\) Hamming code support the Steiner system \(S(2,3,2^r-1)\) [14; pg. 89].
- \([2^r,2^r-r-1,4]\) Extended Hamming code — Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [14; pg. 89].
- \(q\)-ary quadratic-residue (QR) code — The supports of fixed-weight codewords of certain \(q\)-ary QR codes support combinatorial designs [15–17].
- Pless symmetry code — The supports of fixed-weight codewords of certain Pless symmetry codes support combinatorial designs [17–19].
- Golay code — The supports of the weight-seven (weight-eight) codewords of the (extended) Golay code support the Steiner system \(S(4,7,23)\) (\(S(5,6,12)\)) [17,20][14; pg. 89]. Its blocks are called octads.
- Ternary Golay code — The supports of the weight-five (weight-six) codewords of the (extended) ternary Golay code support the Steiner system \(S(4,5,11)\) (\(S(5,6,12)\)) [17,20][14; pg. 89]. Its blocks are called hexads.
- Perfect code — Perfect codes and combinatorial designs are related [15,21].
- Dual linear code — Linear codes and their duals are related to combinatorial designs via the Assmus-Mattson theorem [16,22] (see [7; Sec. 5.4]).
- Self-dual linear code — Self-dual extremal codes yield combinatorial \(\leq 5\)-designs using the Assmus-Mattson theorem [16] (see [7; Sec. 5.4]). See [23; Table 1.61, pg. 683] for a table of combinatorial designs obtained from self-dual codes.
- Gallager (GL) code — Some Steiner systems can be used to construct Gallager codes [24].
- Algebraic LDPC code — Combinatorial designs can be used to construct explicit LDPC codes [25–27].
- Hadamard code — Hadamard designs are combinatorial designs constructed from Hadamard matrices [28]; see Ref. [8].
- \([48,24,12]\) self-dual code — Fixed-weight codewords of extremal self-dual doubly-even codes whose length divides 24 form a combinatorial 5-design [16].
- Higman-Sims graph-adjacency code — Codewords of weight 36 of the Higman-Sims graph-adjacency code form a \(2\)-\((100,36,525)\) design [29; 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 [29; Prop. 1.1].
- Preparata code — Preparata codewords of each weight form a 3-design [11; pg. 471].
- Nordstrom-Robinson (NR) code — NR codewords give \(3\)-\((16, 6, 4)\), \(3\)-\((16, 8, 3)\), and \(3\)-\((16, 10, 24)\) designs [11; pg. 164].
- Mixed code — Combinatorial designs have been generalized to mixed alphabets [30].
- Dodecacode — There exists a \(5\)-\((12, 6, 3)\) design in the dodecacode, and a \(3\)-\((11, 5, 4)\) design in the shortened dodecacode [31].
- Editing code — Perfect deletion correcting codes can be constructed using combinatorial design theory [32,33].
- Perfect-tensor code — Combinatorial designs and \(d\)-uniform quantum states are related [34].
- EA combinatorial-design QLDPC code — Combinatorial designs can be used to construct EA QLDPC codes [35].
- Jump code — Certain types of combinatorial designs can be used to obtain jump codes [36–38].

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## Page edit log

- Victor V. Albert (2023-04-24) — most recent

## Cite as:

“Combinatorial design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/combinatorial_design