42 lines on a chessboard with associated numbers

Question

The 64 squares of a chessboard can be associated with 42 lines as follows:
the 8 rows
the 8 columns
13 diagonals from north-west to south-east
13 diagonals from north-east to south-west
Those diagonals cover 2,3,4,5,6,7 or 8 squares of the same color. The squares at the corner don't build a diagonal.

Now 40 squares out of the 64 squares are selected and a token is placed on each of the 40 squares.
Each of the 42 lines, which have an even number of tokens, counts as one point.
What is the maximum number of points, which can be reached?

Answer 1

I can get

The full 42 points

As we have a multiple of 8 tokens, it is possible to get all lines even by using a pattern with 8-fold symmetry that avoids the main diagonals: E.g.

 0 1 1 1 1 1 1 0
 1 0 1 1 1 1 0 1
 1 1 0 0 0 0 1 1
 1 1 0 0 0 0 1 1
 1 1 0 0 0 0 1 1
 1 1 0 0 0 0 1 1
 1 0 1 1 1 1 0 1
 0 1 1 1 1 1 1 0