Give a high-level description of a function f which takes as input a number n N and returns the descriptions of two Turing machines, C_n and D_n, that work as follows. C_n will be a compressor for strings of exactly length n, and D_n will be a decompressor for strings of exactly length n. (They can be assumed to only work for strings of length n.) When C_n is given as input a string s {0, 1}ⁿ (i.e. a binary string of length n) it returns a binary string c {0, 1}^m such that for every input string of length n except one, m < n. That is, it returns a binary string shorter than length n for every possible input of length n except for one. For the one exception, the c returned is of length n. Then, when c is used as input to D_n, D_n outputs the original string s (that was input to C_n). Note that m can also be 0, i.e. c can be the empty string. Given some output c, which was output from C_n on input s, Dn must always correctly output that s.

In summary, C_n is a compressor for strings of exactly length n and D_n is a decompressor for those strings. C_n and D_n are hard-coded to work for strings for length n, and compress and decompress all but one of them by at least one bit.