Every Binary (2m-2, 22m-2-m, 3) Code Can Be Lengthened to Form a Perfect Code of Length 2m-1

We answer a problem posed by Etzion and Vardy (1998) by showing that a binary code of length N=2(m)-2 with 2(N-m) codewords and minimum distance three can always be lengthened to form a perfect code of length 2(m-1).