SIAM Journal on Discrete Mathematics
On Perfect Codes and Tilings: Problems and Solutions
SIAM Journal on Discrete Mathematics
On Perfect Codes: Rank and Kernel
Designs, Codes and Cryptography
Full-Rank Tilings of $\mathbbF^8_\!2$ Do Not Exist
SIAM Journal on Discrete Mathematics
On the Ranks and Kernels Problem for Perfect Codes
Problems of Information Transmission
Resolving the Existence of Full-Rank Tilings of Binary Hamming Spaces
SIAM Journal on Discrete Mathematics
A Full Rank Perfect Code of Length 31
Designs, Codes and Cryptography
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The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C驴 are linearly equivalent if there exists a non-singular matrix A such that AC = C驴 where C and C驴 are matrices with the code words of C and C驴 as columns. Hessler proved that the perfect codes C and C驴 are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes.