Quadtree encoding (a simple compression algorithm) (noch nicht übersetzt)

The quadtree encoding allows us to describe a 2^N×2^N black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:

• the first bit deals with the complete 2^N×2^N region;
• "0" denotes a split:
  the current 2ⁿ×2ⁿ region is divided into 4 sub-regions of dimension 2^n-1×2^n-1,
  the next bits contains the description of the top left, top right, bottom left and bottom right sub-regions - in that order;
• "10" indicates that the current region contains only black pixels;
• "11" indicates that the current region contains only white pixels.

Consider the following 4×4 image (colored marks denote places where a split can occur):

This image can be described by several sequences, for example : "001010101001011111011010101010", of length 30, or
"0100101111101110", of length 16, which is the minimal sequence for this image.

For a positive integer N, define D_N as the 2^N×2^N image with the following coloring scheme:

• the pixel with coordinates x = 0, y = 0 corresponds to the bottom left pixel,
• if (x - 2^N-1)² + (y - 2^N-1)² ≤ 2^2N-2 then the pixel is black,
• otherwise the pixel is white.

What is the length of the minimal sequence describing D₂₄ ?