On the Characterization of Linear Uniquely Decodable Codes
Designs, Codes and Cryptography - Special issue on designs and codes—a memorial tribute to Ed Assmus
On the classification of perfect codes: side class structures
Designs, Codes and Cryptography
The perfect binary one-error-correcting codes of length 15: part II-properties
IEEE Transactions on Information Theory
Optimal hash functions for approximate matches on the n-cube
IEEE Transactions on Information Theory
SIAM Journal on Discrete Mathematics
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We study partitions of the space $\F{n}$ of all the binary $n$-tuples into disjoint sets, where each set is an additive coset of a given set $V$. Such a partition is called a tiling of $\F{n}$ and denoted $(V,A)$, where $A$ is the set of coset representatives. We give a sufficient condition for a set $V$ to be a tile in terms of the cardinality of $V\!+\!V$. We then employ this condition to classify all tilings with sets of small cardinality. Further, periodicity of tilings in $\F{n}$ is discussed, and a simple construction of nonperiodic tilings of $\F{n}$ is presented for all $n \ge 6$. It is also shown that the nonperiodic tiling of $\F{6}$ is unique. A tiling $(V,A)$ is said to be proper if $V$ generates $\F{n}$; it is said to be full rank if both $V$ and $A$ generate $\F{n}$. We show that, in general, the classification of tilings can be reduced to the study of proper tilings. We then prove that any %proper tiling may be decomposed into smaller tilings that are either trivial or have full rank. Existence of full-rank tilings is exhibited by showing that each tiling is uniquely associated with a perfect binary code. Moreover, it is shown that periodic full-rank tilings may be further decomposed into smaller tilings, and then the existence of nonperiodic full-rank tilings is deduced. Finally, we generalize the well-known Lloyd theorem, originally stated for tilings by spheres, for the case of arbitrary tilings.