ON PERFECT CODES AND TILINGS - PROBLEMS AND SOLUTIONS

Although nontrivial perfect binary codes exist only for length n = 2(m ) - 1 with m greater than or equal to 3 and for length n = 23, many pr oblems concerning these codes remain unsolved. Herein, we present solu tions to some of these problems. In particular, we show that the small est nonempty intersection of two perfect codes of length 2(m) - 1 cons ists of two codewords, for all m greater than or equal to 3. We also p rovide a complete solution to the intersection number problem for Hamm ing codes. Furthermore, 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. This result implies the existence of distinct g eneralized Hamming weights for perfect codes, and we determine complet ely the generalized Hamming weights of all perfect codes that do not c ontain embedded full-rank perfect codes. We further explore the close ties between perfect codes and tilings: we prove that full-rank tiling s of F-2(n) exist for all n greater than or equal to 14 and show that the existence of full-rank tilings for other n is closely related to t he existence of full-rank perfect codes with kernels of high dimension . We briefly survey the present state of knowledge on perfect binary c odes and list several interesting and important open problems concerni ng perfect codes and tilings.