hi ArtGraf! here is what i would do (sorry - the script i recorded is going astray at one point, so heres a written step by step):
draw default circle like you have done
selection submenu: mask all; sel=1; skp=1; apply row
colour: fill with white
selection: inverse
colour: fill with red
selection: clear
okay, looks like yours when pf is off, right?
while the bullseye is still active, go to texture submenu in tool modifiers and press col>tex - this will create a texture in the texture palette. hover with the cursor above it to read its size. it was 16 by 8 in my case. in case the same is valid for you, clear and resize the canvas to a 2:1 ratio (much bigger than the created texture, like 600:300 for example). select flat material. fill canvas with texture. choose the mrgbz-grabber and grab all of the canvas to create a bigger (hi-res) texture. clear canvas. recreate your circle and apply new (bigger) texture - voila 
hope this helps! there might be easier ways but thats all i could think of for now 