# 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.

```magma
> 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

```