On Perfect Codes: Rank and Kernel
Designs, Codes and Cryptography
On the Ranks and Kernels Problem for Perfect Codes
Problems of Information Transmission
The Classification of Some Perfect Codes
Designs, Codes and Cryptography
Kernels and p-Kernels of pr-ary 1-Perfect Codes
Designs, Codes and Cryptography
A Full Rank Perfect Code of Length 31
Designs, Codes and Cryptography
On Intersection Problem for Perfect Binary Codes
Designs, Codes and Cryptography
On the classification of perfect codes: side class structures
Designs, Codes and Cryptography
Maximal partial packings of $${Z_2^n}$$ with perfect codes
Designs, Codes and Cryptography
Discrete Applied Mathematics
On transitive partitions of an n-cube into codes
Problems of Information Transmission
On the intersection of Z2Z4-additive Hadamard codes
IEEE Transactions on Information Theory
The perfect binary one-error-correcting codes of length 15: part II-properties
IEEE Transactions on Information Theory
Two optimal one-error-correcting codes of length 13 that are not doubly shortened perfect codes
Designs, Codes and Cryptography
Problems of Information Transmission
SIAM Journal on Discrete Mathematics
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Although nontrivial perfect binary codes exist only for length n = 2m -1 with $m \ge 3$ and for length n=23, many interesting 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 $m \ge 3$. We also provide a complete solution to the intersection number problem for Hamming codes. Furthermore, we prove that for $m \ge 3$, a perfect code of length 2m-1 -1 is embedded in a perfect code $\Bbb{C}$ of length 2m -1 if and only if ${\Bbb C}$ 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 ${\Bbb F}_{2}^n$ exist for all $n \geq 14$ 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.