On the classification of perfect codes: side class structures
Designs, Codes and Cryptography
SIAM Journal on Discrete Mathematics
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We show that there are no full-rank tilings of $\mathbb{F}^8_{\kern -1pt 2}$, using a carefully designed exhaustive search. This solves an open problem posed in [T. Etzion and A. Vardy, SIAM J. Discrete Math., 11 (1998), pp. 205--233]. This also implies that a full-rank perfect binary code of length 15 with a kernel of dimension 7 does not exist.