I'm with you, mooncow. Since I have a background in IT, I thought that "Basic" in the title indicated a numerical base: binary is base 2, decimal is base 10, hexadecimal is base 16... This was reinforced by finding lots of occurrences of ONE in contiguous cells, and even a TWO!
However, after a doddle of a grid fill and a simple one-sentence explanation of the endgame, no amount of staring helped and I threw the puzzle to one side in frustration until today. Still no joy, so I came here.
Thanks, malone (#24), for putting me on the right track without explicitly revealing the theme: sadly not something of which I have a great deal of knowledge!
Now I'm left with only one query: in 13ac, how does "Nut" supply the two letters required in the solution?