Associated \(3\)-Designs

HadamardRowDesign(H, i): AlgMatElt, RngIntElt -> Dsgn

Given an \(n \times n\) Hadamard matrix \(H\) (with \(n \ge 4\)) and an integer \(i\) with \(1 \le i \le n\), returns the Hadamard \(3\)–design corresponding to the \(i\)th row of \(H\).

HadamardColumnDesign(H, i): AlgMatElt, RngIntElt -> Dsgn

Given an \(n \times n\) Hadamard matrix \(H\) (with \(n \ge 4\)) and an integer \(i\) with \(1 \le i \le n\), returns the Hadamard \(3\)–design corresponding to the \(i\)th column of \(H\).

Example: Hadamard Designs (ex-16b02c)

There is only one Hadamard equivalence class of \(8 \times 8\) Hadamard matrices. We construct one matrix, and two of its designs.

> R := MatrixRing(Integers(), 8);
> H := R![1,  1,  1,  1,  1,  1,  1,  1,
>         1,  1,  1,  1, -1, -1, -1, -1,
>         1,  1, -1, -1,  1,  1, -1, -1,
>         1,  1, -1, -1, -1, -1,  1,  1,
>         1, -1,  1, -1,  1, -1, -1,  1,
>         1, -1,  1, -1, -1,  1,  1, -1,
>         1, -1, -1,  1, -1,  1, -1,  1,
>         1, -1, -1,  1,  1, -1,  1, -1];
> IsHadamard(H);
true
> DR := HadamardRowDesign(H, 3);
> DR: Maximal;
3-(8, 4, 1) Design with 14 blocks
Points: {@ 1, 2, 3, 4, 5, 6, 7, 8 @}
Blocks:
    {1, 2, 5, 6},
    {1, 2, 7, 8},
    {1, 2, 3, 4},
    {1, 4, 5, 7},
    {1, 4, 6, 8},
    {1, 3, 6, 7},
    {1, 3, 5, 8},
    {3, 4, 7, 8},
    {3, 4, 5, 6},
    {5, 6, 7, 8},
    {2, 3, 6, 8},
    {2, 3, 5, 7},
    {2, 4, 5, 8},
    {2, 4, 6, 7}
> HadamardColumnDesign(H, 8);
3-(8, 4, 1) Design with 14 blocks

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