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Challenges

# Decoding a non injective bit matrix encoding

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### 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 1s in each line (top-to-bottom)
• S integers corresponding to the number of 1s in each column (left-to-right)
• 2 integers corresponding to the number of 1s in each diagonal (main, then anti-diagonal)
• 4 integers corresponding to the number of 1s 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 0s and 1s

### Example

#### Encoding

4
1 1 1 1
1 1 1 1
0 4
0 2 2 0
1 2 2 1
1 2 2 1


#### Decoded matrix

1
0001
0010
0100
1000


### Scoring

The goal for this challenge is the produce the fastest algorithm (i.e, the algorithm with the smallest asymptotic complexity), and as such you should include an short analysis of your algorithm alongside your code.

Despite that, all solutions will be evaluated by me on the same machine, and the time measurements posted as a comment on the corresponding answer. This is merely for curiosity, not for actual scoring.
You can expect the latest versions for each language and compiler/interpreter. For languages with JITs and similar tools, those will be used (e.g for Python, PyPy will be used).

If you manage to get your decoder to run in under a second for the bigger cases, you will beat me :D

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

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