I'm a doctor studying tiny specimen under a microscope. I had this strange idea to "barcode" each specimen with a physical barcode. This barcode is assembled from two pigments which can be analogized as {1, 0}.
My goal is to find an error-correcting barcode that can be physically assembled from these two pigments, and decodable even despite noisy images (hence, the importance of error-correcting bits).
Because the microscope resolution is finite, there's a certain "agglomeration" size of pigments that can be visualized. For instance, a codeword with many alternating bits (ie. 10101010) blurs together and cannot be distinguished. But, a codeword with agglomeration >= $a$ (ie. 11110001110000 has $a$ = 3) can be distinguished.
My question is, for some linear binary code of length $n$ and rank $k$, what is the subspace of $ \mathbb {F} _{2}^{n}$ where the shortest stretch of any digit is >= $a$ called? Are there robust ways of discovering all such codewords? (Is this related by any chance to covering sets?). I realize this is a bit more applied than most questions here, please also forgive my ignorance!