Binary Strings and Graphs

Binary strings of length n can be added and permuted. Of special interest is the permutation that cyclicly shifts a binary string to the right. For example, when n = 5, then cyclicly shifting 10110 to the right yields 01011. Obviously if we cyclicly shift a string of length n to the right n times to get back what we started with. Addition of the strings is done component-wise modulo 2. For instance, when n = 7, then 1011000+1100101 = 0111101. We spent the semester studying the operation that cyclicly shifts a string to the right once, and then adds the result back to the original. Here are some examples for when n = 4: 1010→ 1111, 1000→ 1100 and 0111→ 1100.