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~~Decoding a non injective bit matrix encoding~~

Decoding a non injective bit matrix encoding [FINALIZED]

### The problem
Someone has created an encoding format for square bit matrices, however they have found it isn't perfect! One encoding may not decode to exactly one matrix, or it may not even be possible to decode.
Knowing this, you're tasked to write a program to decode a set of encodings and since time is of the essence, it's asked that you make it as fast as possible.
The encoding is as follows:
* an integer, **S**, indicating the matrix size (2 for 2×2, 3 for 3×3, etc; always ≥2)
* **S** integers corresponding to the number of 0s in each line (top-to-bottom)
* **S** integers corresponding to the number of 0s in each column (left-to-right)
* 2 integers corresponding to the number of 0s in each diagonal (main, then anti-diagonal)
* 4 integers corresponding to the number of 0s in each quadrant (top-right, top-right, bottom-left, bottom-right)
* **S** integers corresponding to the number of transitions in each line (top-to-bottom)
* **S** integers corresponding to the number of transitions in each column (left-to-right)
The quadrants are defined by the side length divided by two, floored.
Here's an example of a 5×5 matrix, with quadrant boundaries highlighted by different digits:
11222
11222
33444
33444
33444
The program should output the number of matrices possible to decode, followed by a representation of such matrices.
That representation should be composed of `0`s and `1`s
### Example
#### Encoding
4
1 1 1 1
1 1 1 1
0 2
0 2 2 0
1 2 2 1
1 2 2 1
#### Decoded matrix
1
0001
0010
0100
1000
### Time constraints
The upper bound for evaluation is **20 seconds** to allow everyone to use any language they want.
If you manage to get your decoder to run in **under a second** for the bigger cases, you will beat me!
### Evaluation
All solutions will be evaluated by me on the same machine, and the time measurements posted as a comment on the corresponding answer.
You can expect the latest versions for each language and compiler/interpreter. For Python, PyPy will be used, to make it a more interesting option.
There will be some extra hidden test cases.
---
### More test cases
#### Input 1
4
2 3 4 3
2 3 4 3
2 4
4 3 4 1
1 1 0 1
1 1 0 1
#### Output 1
0
#### Input 2
3
0 2 0
1 0 1
0 0
0 0 1 1
0 2 0
2 0 2
#### Output 2
1
000
101
000
---
### Scoreboard
(no submissions yet)

## Now posted: [Decoding a non injective bit matrix encoding](https://codegolf.codidact.com/posts/287925 "View the finished challenge")
---
### The problem
Someone has created an encoding format for square bit matrices, however they have found it isn't perfect! One encoding may not decode to exactly one matrix, or it may not even be possible to decode.
Knowing this, you're tasked to write a program to decode a set of encodings and since time is of the essence, it's asked that you make it as fast as possible.
The encoding is as follows:
* an integer, **S**, indicating the matrix size (2 for 2×2, 3 for 3×3, etc; always ≥2)
* **S** integers corresponding to the number of 0s in each line (top-to-bottom)
* **S** integers corresponding to the number of 0s in each column (left-to-right)
* 2 integers corresponding to the number of 0s in each diagonal (main, then anti-diagonal)
* 4 integers corresponding to the number of 0s in each quadrant (top-right, top-right, bottom-left, bottom-right)
* **S** integers corresponding to the number of transitions in each line (top-to-bottom)
* **S** integers corresponding to the number of transitions in each column (left-to-right)
The quadrants are defined by the side length divided by two, floored.
Here's an example of a 5×5 matrix, with quadrant boundaries highlighted by different digits:
11222
11222
33444
33444
33444
The program should output the number of matrices possible to decode, followed by a representation of such matrices.
That representation should be composed of `0`s and `1`s
### Example
#### Encoding
4
1 1 1 1
1 1 1 1
0 2
0 2 2 0
1 2 2 1
1 2 2 1
#### Decoded matrix
1
0001
0010
0100
1000
### Time constraints
The upper bound for evaluation is **20 seconds** to allow everyone to use any language they want.
If you manage to get your decoder to run in **under a second** for the bigger cases, you will beat me!
### Evaluation
All solutions will be evaluated by me on the same machine, and the time measurements posted as a comment on the corresponding answer.
You can expect the latest versions for each language and compiler/interpreter. For Python, PyPy will be used, to make it a more interesting option.
There will be some extra hidden test cases.
---
### More test cases
#### Input 1
4
2 3 4 3
2 3 4 3
2 4
4 3 4 1
1 1 0 1
1 1 0 1
#### Output 1
0
#### Input 2
3
0 2 0
1 0 1
0 0
0 0 1 1
0 2 0
2 0 2
#### Output 2
1
000
101
000
---
### Scoreboard
(no submissions yet)

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