On perfect codes and tilings: problems and solutions

Although nontrivial perfect binary codes exist only for lengths n=2^m-1 and n=23, many problems concerning these codes remain unsolved. We present solutions to some of these problems. In particular, we show that the smallest nonempty intersection of two perfect codes of length 2^m-1 consists of two codewords, for all m⩾3. We prove that a perfect code of length 2^m-1-1 is embedded in a perfect code C of length 2^m-1, if and only if C is not of full rank. Using this result, we determine the generalized Hamming weight hierarchy of most perfect codes. We further explore the close ties between perfect codes and tilings: we prove that full-rank tilings of F₂ⁿ exist for all n⩾14, and show that the existence of full-rank tilings for other n is closely related to the existence of full-rank perfect codes with large kernels