Dont you need to explain a little more about what 'close' means? As far as I understand, DNA consists of CGTA or some 4 letters like that. How can the given 15 character string be 'close' to a 15 character sequence in the large DNA file? Maybe some examples (link to data?) would be a good idea to show us what the result should be.
I would think that any approach will need to compare the given 15 characters with each sliding window of 15 characters in the large file, but that is just an uneducated opinion as I am not a maths guy. I think this kind of thing is going to be fairly slow because you have to test each 15-character window whilst moving the window along by one character each time. I am sure someone else probably has a better idea than me how to do this.
I would think that any approach will need to compare the given 15 characters with each sliding window of 15 characters in the large file, but that is just an uneducated opinion as I am not a maths guy. I think this kind of thing is going to be fairly slow because you have to test each 15-character window whilst moving the window along by one character each time. I am sure someone else probably has a better idea than me how to do this.