this will
- recode for $\#steps \in [10, 20, 100]$
- at $t = 0$, $k$ random shards among the $n$ encoded will be selected at random
- at $t \geq 1$, all $k$ shards will be used to recode $k$ brand new shards
- make sure the last set of $k$ shards recoded $\#steps$ together can decode the data

## example with $(k, n) = (3, 5)$ and $\#steps = 3$
- $(s_i)_{1 \leq i \leq k}$ are the $k$ source shards
- $(e_j)_{1 \leq j \leq n}$ are the $n$ encoded shards
- $(m_i)_{1 \leq i \leq k}$ are the $k$ randomly selected shards
- $(n_i)_{1 \leq i \leq k}$ are the shards after step $1$
- $(o_i)_{1 \leq i \leq k}$ are the shards after step $2$
- $(p_i)_{1 \leq i \leq k}$ are the shards after step $3$
- the $(p_i)_{1 \leq i \leq k}$ will be used for decoding

```mermaid
graph TD;

    s1 --> e1; s1 --> e2; s1 --> e3; s1 --> e4; s1 --> e5;
    s2 --> e1; s2 --> e2; s2 --> e3; s2 --> e4; s2 --> e5;
    s3 --> e1; s3 --> e2; s3 --> e3; s3 --> e4; s3 --> e5;

    e1 --> m1;
    e3 --> m2;
    e4 --> m3;

    m1 --> n1; m1 --> n2; m1 --> n3;
    m2 --> n1; m2 --> n2; m2 --> n3;
    m3 --> n1; m3 --> n2; m3 --> n3;

    n1 --> o1; n1 --> o2; n1 --> o3;
    n2 --> o1; n2 --> o2; n2 --> o3;
    n3 --> o1; n3 --> o2; n3 --> o3;

    o1 --> p1; o1 --> p2; o1 --> p3;
    o2 --> p1; o2 --> p2; o2 --> p3;
    o3 --> p1; o3 --> p2; o3 --> p3;
```

- pass `n` to `try_all_decoding_combinations` and don't try to decode when shards have been recoded ($\#shards > n$) and there are no recoded shards in the $k$ combination under review ($\max(is) < n$)
- pass `recoding_steps` and `should_not_be_decodable` as arguments to `end_to_end_with_recoding_template`
- fix $n = 5$ => this leads to tests that run in less than 10sec again
- add $(k, n) = (8, 10)$ => tests still run in less than 13sec
- split recoding scenarii into "_simple_" and "_chain_"
- show indices in a "_pretty_" format, i.e. showing indices greater than $n$ as `(n)`, `(n + 1)`, ...

- `komodo::linalg::Matrix::random` is tested
- `komodo::linalg::Matrix::inverse` is tested on more matrix sizes, from $1$ to $20$ random matrices
- `komodo::field` tests have been double-checked
- pure "recoding" tests from `komodo::fec` have been double-checked
- `end_to_end` and `end_to_end_with_recoding` now runs for $k \in [3, 5]$ and $\rho \in [\frac{1}{2}, \frac{1}{3}]$ with $n = \lfloor \frac{k}{\rho} \rfloor$
- all "_$k$ among $n + t$_" combinations are tested with `try_all_decoding_combinations`, possibly with some removals in case recoding is involved with `is_inside`

> ❗ **Important**  
> on my machine, `make test` goes from less than 8sec on latest `main` to around 40sec with this MR

they have been moved to [dragoon/binaries](https://gitlab.isae-supaero.fr/dragoon/binaries).

this is the trivial case where it is always possible to decode the original data, so there is no need to have it here.

this MR
- moves the Nushell modules from `bins/inbreeding/` to `bins/inbreeding/src/.nushell/`
- creates a `NUSHELL` constant in `consts.nu` to allow the following more robust construct
```bash
use consts.nu
use $consts.NUSHELL ...
```
- updates the README

this simply makes "parsing" in the inbreeding modules more robust and centralized.

- early return when format is bad twice
- refactor

wait for
- !134 

## description
assuming !134 has been merged, this MR allows to run something like
```bash
inbreeding list | input list --fuzzy | inbreeding load $in | inbreeding plot
```
which will fuzzy find the available experiments and then plot it after load 👌

this can be used by `plot` without passing extra arguments, i.e. the pipeline becomes
```bash
use bins/inbreeding

let experiment = "..."
```
```bash
inbreeding load $experiment | inbreeding plot
```
instead of having to know the value of `k` and do
```bash
inbreeding load $experiment | inbreeding plot --options { k: $k }
```

- addresses #9 
- needed !133 to work

this MR simply adds `+ Clone` to `rng` and removes the `mut` from the `rng` of the `measure_inbreeding` function.

running the same snippet of code from #9 yields the following two images with a schedule of $1$ and $5$ respectively

![000_1.1](/uploads/ace74e299c6d4374df713ab5004f3c1d/000_1.1.png)
![000_1.2](/uploads/c784f5b62c41004971fd3048f7b5babc/000_1.2.png)

we see that all measurements on $t$ where $t = 0 \mod 5$ are the same in both images 🎉

related to
- #9

in order for the measurements not to influence the experiment, the seeds passed to the runs need to not include the _measurement schedule_ parameters!

EDIT: in the end, it's more than that, we want to only include things related to the environment in its hash, nothing related to the measurements, i.e. we want to either
- exclude `strategies`, `nb_scenarii`, `measurement_schedule`, `measurement_schedule_start`, `nb_measurements` and `max_t`
- include `nb_bytes`, `k`, `n` and `environment`

## output sample
let's say the current seed is `"b239e48345ac457b492cc164f58c010d07292c88e4791e607d91796baec7f334"` and the experiment has ID `fixed:0-single:3-5-50-10240`, and that we are watching both the creation of the first experiment and a few runs, e.g.
- `002d8de28913efbf7dbd111b817ae901fee2d47882ba7aa76d293c2d95d9652c`
- `015672b37c9cf1a6b475937987294f9a503a922ffbcfdfc5d18ef839fac91b8c`
- `080a51a17ac43fcbdf08f77f3bc993983fef589bd9672f04382af4b16dd09b13`

then the output would look like
```
b239e48  fixed:0-single:3-5-50-10240            at 2024-05-30 15:24:12
b239e48  fixed:0-single:3-5-50-10240  002d8de   at 2024-05-30 15:24:14
b239e48  fixed:0-single:3-5-50-10240  015672b   at 2024-05-30 15:24:16
b239e48  fixed:0-single:3-5-50-10240  080a51a   at 2024-05-30 15:24:18
```

we are switching
- _naive recoding_ to _$(k, 1)$-re-encoding_
- _true recoding_ to _$k$-recoding_

this is to have a complete identifier for each run, which does not depend at all on the time at which the experiment runs.

related to
- !122

## description
!122 introduced a `draw_unique_indices` function which uses a `HashSet` to accumulate unique indices in the range `0..<len`.
however, a `HashSet` does not preserve the order of insertion when iterating over the elements of the set... which results in apparent randomness, even though the RNG seed is the same 😮 

this MR switches back to using `shuffle` which used to work, even though a bit less performant 👌   
it's basically a revert of !122, while keeping the refactoring into `random.rs`.

## measuring the performance
i did run the same timing experiment from !122 but with `main` on `bb55005f` and the MR on `fix-shuffle`

| env     | main                   | mr                      | improvement         |
| ------- | ---------------------- | ----------------------- | ------------------- |
| fixed:0 | 6sec 244ms 238µs 45ns | 8sec 734ms 929µs 328ns | -39.88783363238997  |
| fixed:1 | 639ms 720µs 39ns      | 731ms 360µs 261ns      | -14.325051024390373 |

we loose a bit

- add optional `$.help` to argument `err` of `error throw`
- parse `prng_seed: [u8; 32]` in `rng` and `inbreeding`
- compute the "_local_" seed by hashing the "_global_" seed, the strategy and the iteration index
- pass `--prng-seed: string`, a 64-char long seed to `inbreeding run`

because the "_environment_" can contain an experiment separator, e.g. `random-fixed`, more powerful `parse` patterns needs to be used.

* 7b44a886 refactor constants from `inbreeding load`
* 64e446f7 move `remove-cache-prefix` to new `path.nu`
* 822b03fd add `inbreeding inspect`

in order to add more information in the experiment names, e.g. $k$, $n$ and $\#\text{bytes}$, this MR changes the cache files format from
```
{seed}/{timestamp}/{env}/{strategy}/...
```
to
```
{seed}/{timestamp}-{env}-{strategy}-{k}-{n}-{nb_bytes}/...
```

this MR turns `./.nushell/` into a directory module by
- adding `mod.nu`
- exporting all the modules

all uses of `.nushell/` have been fixed to not mention `.nu` internal modules anymore.

> 💡 **Note**  
> the `.nushell venv` module has been removed because, when the `$venv.VENV` activation script is not there, Nushell can't parse the whole `.nushell` module, which is very annoying to have to rely of the state of the external filesystem to be able to simply parse a module...

this also bumps Nushell to 0.93.0 to include the "extra" command `fmt`, see [Nushell 0.92.0](https://www.nushell.sh/blog/2024-04-02-nushell_0_92_0.html#incorporating-the-extra-feature-by-default-toc)

in `./bins/inbreeding/`, this MR does
- refactor the "list item drawing" from `environment.rs` and `strategy.rs` into the `draw_unique_elements` function of new `random.rs` module
- use a `HashSet` to draw unique indices in the slice of "things" to draw from and then extracts the items corresponding to these indices

## results
```bash
use ./bins/inbreeding
use std bench

const PRNG_SEED = 0
const OPTS = {
    nb_bytes: (10 * 1_024),
    k: 10,
    n: 20,
    nb_measurements: 100,
    nb_scenarii: 10,
    measurement_schedule: 1,
    measurement_schedule_start: 2_000,
    max_t: 2_000,
    strategies: [ "single:5" ],
    environment: null,
}

def run [rev: string] {
    git co $rev

    inbreeding build

    let a = bench --rounds 5 {
        inbreeding run --options ($OPTS | update environment "fixed:0") --prng-seed $PRNG_SEED
    }
    let b = bench --rounds 5 {
        inbreeding run --options ($OPTS | update environment "fixed:1") --prng-seed $PRNG_SEED
    }

    {
        0: $a,
        1: $b,
    }
}

let main = run a29b511d
let mr = run fix-shuffle
```
```bash
let table = [
    [env, main, mr, improvement];

    ["fixed:0", $main."0".mean, $mr."0".mean, (($main."0".mean - $mr."0".mean) / $main."0".mean * 100)],
    ["fixed:1", $main."1".mean, $mr."1".mean, (($main."1".mean - $mr."1".mean) / $main."1".mean * 100)],
]

$table | to md --pretty
```

| env     | main                    | mr                      | improvement        |
| ------- | ----------------------- | ----------------------- | ------------------ |
| fixed:0 | 8sec 504ms 794µs 784ns | 6sec 353ms 206µs 645ns | 25.298530930431912 |
| fixed:1 | 727ms 648µs 292ns      | 639ms 443µs 782ns      | 12.12186037811795  |

the improvement is quite nice, even though not huge, but the code is cleaner anyways 🙏

this adds `$.diversity.e` to the output of `inbreeding load` and input of `inbreeding plot` to show errors bars in the final plot.

these can be discarded by running something similar to
```bash
inbreeding load $experiment
    | reject diversity.e
    | inbreeding plot --options { k: $OPTS.k }
```

plotting "inbreeding" results is now done in two steps
- `load` the data from raw experiment files with `inbreeding load`
- `plot` the results by piping this data into `inbreeding plot`

## new structure for the repository
- benchmarks are in `./benchmarks/` and can be run with either `cargo run --package benchmarks --bin <bench>` or the commands in `./benchmarks/README.md`
```
├── Cargo.toml
├── README.md
└── src
    └── bin
        ├── commit.rs
        ├── fec.rs
        ├── linalg.rs
        ├── operations
        │   ├── curve_group.rs
        │   └── field.rs
        ├── recoding.rs
        ├── setup.rs
        └── setup_size.rs
```

- examples are now in `./bins/` as standalone binaries and can be run either with `cargo run --package <pkg>` or with the help of the `cargo bin` command from `.nushell/cargo.nu`
```
├── curves
│   ├── Cargo.toml
│   ├── README.md
│   └── src
│       └── main.rs
├── inbreeding
│   ├── build.nu
│   ├── Cargo.toml
│   ├── consts.nu
│   ├── mod.nu
│   ├── plot.nu
│   ├── README.md
│   ├── run.nu
│   └── src
│       ├── environment.rs
│       ├── main.rs
│       └── strategy.rs
├── rank
│   ├── Cargo.toml
│   └── src
│       └── main.rs
└── rng
    ├── Cargo.toml
    └── src
        └── main.rs
```

- Nushell modules are now located in `./.nushell/`

## changelog
apart from the changes to the general structure of the repo:
- `binary.nu` -> `.nushell/binary.nu`
- new `cargo bin` command from `.nushell/cargo.nu`
- `error throw` is now defined in `.nushell/error.nu`
- main TOML has been greatly simplified because the dependencies of "examples" have been moved to the associated crates
- the rest is basically the same but in the new structure

related to
- dragoon/komodo!113

this MR makes sure that the seeds given to each "strategy + scenario" loop are different by generating a bunch of unique seeds per strategy.

this will make `inbreeding run` dump all the data as-is to files on disk.
data will be arranged in the following way
- the `$CACHE` is `~/.cache/komodo/inbreeding/`
- each seed will have its own directory: `$CACHE/<seed>/`
- each strategy and environment will have their own directory: `$CACHE/<seed>/<date>/<env>/<strat>/`
- finally each recoding scenario will go to its own subdirectory: `$CACHE/<seed>/<date>/<env>/<strat>/<i>`

> ❗ **Important**  
> the plotting pipeline is broken for now but will be fixed in later MRs soon...

- add `--prng-seed: u8` to fix the random number generator seed

## example
by running the following snippet, we get
- `first.123.png` and `second.123.png` with `--prng-seed 123` which are the same
- `first.111.png` and `second.111.png` with `--prng-seed 111` which are the same
- `first.111.png` and `first.123.png` are different

```bash
use ./scripts/inbreeding

const OPTS = {
    nb_bytes: (10 * 1_024),
    k: 10,
    n: 20,
    nb_scenarii: 10,
    nb_measurements: 10,
    measurement_schedule: 1,
    measurement_schedule_start: 0,
    max_t: 50,
    strategies: [
        "single:1",
        "double:0.5:1:2",
        "single:2"
        "double:0.5:2:3",
        "single:3"
        "single:5"
        "single:10",
    ],
    environment: "random-fixed:0.5:1",
}

inbreeding build

inbreeding run --options $OPTS --prng-seed 123 --output /tmp/first.123.nuon
inbreeding plot /tmp/first.123.nuon --options { k: $OPTS.k } --save /tmp/first.123.png

inbreeding run --options $OPTS --prng-seed 123 --output /tmp/second.123.nuon
inbreeding plot /tmp/second.123.nuon --options { k: $OPTS.k } --save /tmp/second.123.png

inbreeding run --options $OPTS --prng-seed 111 --output /tmp/first.111.nuon
inbreeding plot /tmp/first.111.nuon --options { k: $OPTS.k } --save /tmp/first.111.png

inbreeding run --options $OPTS --prng-seed 111 --output /tmp/second.111.nuon
inbreeding plot /tmp/second.111.nuon --options { k: $OPTS.k } --save /tmp/second.111.png
```

| seed | first | second |
| ---- | ----- | ------ |
| 123  | ![first.123](/uploads/6b09bf94ca7019e200a47e1a53adc533/first.123.png) | ![second.123](/uploads/fa77c1be84d6279f71cbfe3064c83242/second.123.png) |
| 111  | ![first.111](/uploads/bd31a0832825ecbef48178b2e8689a6f/first.111.png) | ![second.111](/uploads/e1c00e0a6765e82388a3c7142847bbab/second.111.png) |

- define `scripts/color.nu` to manipulate RGB colors, especially mix two colors together
- compute the color of _hybrid recoding strategies_ as a weighted sum of the two _simple recoding strategies_ involved, e.g. if the strategy is "10% of the time recode 2 shards and 90% of the time recode 3", then the color of that curve will be 10% the color of the simple strategy recoding 2 shards and 90% the color of the other simple strategy recoding 3 shards
- make the _hybrid_ curves transparent and dashed

## example
![foo](/uploads/71afca07a87fc38e3fa005cebfab4e50/foo.png)

- add a timestamp to all the measurements of the _diversity_ from `inbreeding/mod.rs`
- allow to delay the measurement starts with `--measurement-schedule-start`, to help completing already existing measurements

> ❗ **Important**  
> existing measurement files will have to change shape from
> ```
> table<strategy: string, diversity: list<float>>
> ```
> to
> ```
> table<strategy: string, diversity: table<t: int, diversity: float>>
> ```

makes sure
- "inbreeding" experiment quits when there are less than $k$ shards
- `fec::decode` returns `KomodoError::TooFewShards` when no shards are provided

just a small QoL improvement

this MR is two-fold
- refactor `run.nu` and `plot.nu` from `scripts/inbreeding/` into Nushell modules with `--options` as argument instead of `options.nu` (a7cebb95, 6b72191f and 5f1c4963)
- introduce another level of depth to the measurements (a0e52e95)

> 💡 **Note**  
> in the table below
> - $s$ is the number of recoding scenarii averages together
> - $m$ is the number of measurements per point
> - two iterations of the same experiment are shown side by side for comparison

s   |    m | . | .
:--:|:----:|:-------------------------:|:-------------------------:
1   | 10   | ![inbreeding_1_10.1](/uploads/c593393edb3513c9d77b0fe134c27fd7/inbreeding_1_10.1.png) | ![inbreeding_1_10.2](/uploads/97c85b36833112de51a2b756ade53479/inbreeding_1_10.2.png)
1   | 100  | ![inbreeding_1_100.1](/uploads/af4da1d7cf76ef43fb39c2a3a529b7cd/inbreeding_1_100.1.png) | ![inbreeding_1_100.2](/uploads/e187298709d524437dea503be6ac555f/inbreeding_1_100.2.png)
1   | 1000 | ![inbreeding_1_1000.1](/uploads/394821777baeff9fec589440ba4c554c/inbreeding_1_1000.1.png) | ![inbreeding_1_1000.2](/uploads/1d592b791075f204f8a0ebdd739403dd/inbreeding_1_1000.2.png)
10  | 100  | ![inbreeding_10_100.1](/uploads/3c822e7669e9f0b4d97919e5a3bd4bca/inbreeding_10_100.1.png) | ![inbreeding_10_100.2](/uploads/aa6b54ec64f82ca386dae4e262dcd0b6/inbreeding_10_100.2.png)
100 | 10   | ![inbreeding_100_10.1](/uploads/7ef383143d981717b0ad01dee0359eb0/inbreeding_100_10.1.png) | ![inbreeding_100_10.2](/uploads/ce71a32a7f6e9ba3c4dae563aeefd856/inbreeding_100_10.2.png)
100 | 100  | ![inbreeding_100_100.1](/uploads/e2038273051f959d8be69fef9ba7a493/inbreeding_100_100.1.png) | ![inbreeding_100_100.2](/uploads/0ef30735597e1a6812484d1cac4d34ca/inbreeding_100_100.2.png)

we can see that
- the smaller the $s$, the more different the two figures are on each line -> this is likely due to the fact that, if only one recoding scenario is used, then repeating the same experiment will result in very different results and measurements. Running the same experiment $s$ times and averaging helps reducing the variance along this axis
- the smaller the $m$, the more noisy the measures of each points -> this is simply because, when $m$ is small, the variance of the empirical means measured for each point is higher

## final results
![inbreeding](/uploads/e561f4e4acad8eedbb3ccf1a4666c302/inbreeding.png)
![inbreeding_100_100_1](/uploads/37ab0eacb5159137595579dfbb20250c/inbreeding_100_100_1.png)