Let $C$ be a binary cyclic Hamming code; we add a parity check at the end of every code word and get an extended (not necessarily cyclic) code.
It is possible to show this code can fix any single error, and any double error occurring in two adjacent position, excluding the final (parity) digit.
Is it possible to permutate the order of the columns in the code's parity check matrix, so that the code (well, an equivalent code) will be able to fix any single error and every double errors in adjacent places (including the two pairs containing the last digit)?