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Decoding a non injective bit matrix encoding

Hello!
I have come across this very interesting programming challenge I thought I'd share.
### The problem
You're tasked to decode an encoding of a square bit matrix of certain size, which contains the number of 0 cells per line, column, diagonal (main and anti-diagonal) and quadrant, as well as the number of transitions per line and column. This encoding isn't injective, that is, it doesn't perfectly map one encoding to one matrix. Thus, your decoder should output how many matrices can be decoded from the given input encoding, and in case there's only one, display it.
### Example
#### Encoding
size: 4
line 0s (left to right): 1, 1, 1, 1
columns 0s (left to right): 1, 1, 1, 1
quadrant 0s: top-right=2 top-left=0, bottom-right=2, bottom-left=0
diagonal 0s: main=0, anti=4
line transitions (left to right): 1, 2, 2, 1
column transitions (left to right): 1, 2, 2, 1
#### Decoded matrix
0001
0010
0100
1000
### Constraints
The matrices are always square and 2x2 or bigger. The goal is to make a decoder that processes on or more 25x25 encodings under a second (on fairly modern hardware, I suppose). Below that, fastest code wins.
### Evaluation
~~All solutions will be evaluated by me on the same machine, and the time measurements posted as comment on the corresponding answer.~~

Hello!
I have come across this very interesting programming challenge I thought I'd share.
### The problem
You're tasked to decode an encoding of a square bit matrix of a certain size, which contains the number of 0 cells per line, column, diagonal (main and anti-diagonal) and quadrant, as well as the number of transitions per line and column. This encoding isn't injective, that is, it doesn't perfectly map one encoding to one matrix. Thus, your decoder should output how many matrices can be decoded from the given input encoding, and in case there's only one, display it.
### Example
#### Encoding
size: 4
line 0s (left to right): 1, 1, 1, 1
columns 0s (left to right): 1, 1, 1, 1
quadrant 0s: top-right=2 top-left=0, bottom-right=2, bottom-left=0
diagonal 0s: main=0, anti=4
line transitions (left to right): 1, 2, 2, 1
column transitions (left to right): 1, 2, 2, 1
#### Decoded matrix
0001
0010
0100
1000
### Constraints
The matrices are always square and 2x2 or bigger. The goal is to make a decoder that processes one or more 25x25 encodings in under a second (on fairly modern hardware, I suppose). Below that, fastest code wins.
### Evaluation
All solutions will be evaluated by me on the same machine, and the time measurements posted as a comment on the corresponding answer.

optimization fastest-code encoding code-challenge binary