On Perfect Codes and Tilings: Problems and Solutions

Although nontrivial perfect binary codes exist only for length n = 2m−1 with m ≥ 3 and for length n = 23, many problems concerning these codes remain unsolved. Herein, we present solutions to some of these problems. In particular, we show that the smallest nonempty intersection of two perfect codes of length 2m − 1 consists of two codewords, for all m ≥ 3. We also provide a complete solution to the intersection number problem for Hamming codes. Furthermore, we prove that a perfect code of length 2m−1 − 1 is embedded in a perfect code C of length 2m − 1 if and only if C is not of full rank. This result implies the existence of distinct generalized Hamming weights for perfect codes, and we determine completely the generalized Hamming weights of all perfect codes that do not contain embedded full-rank perfect codes. We further explore the close ties between perfect codes and tilings: we prove that full-rank tilings of F2 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 kernels of high dimension. We briefly survey the present state of knowledge on perfect binary codes and list several interesting and important open problems concerning perfect codes and tilings.