arXiv subjects

Gabriella Akemi Miyamoto

Publications and source records attributed to Gabriella Akemi Miyamoto.

3 recordsLinked to original sources

arXiv2019

Obtaining binary perfect codes out of tilings

A tiling of the $n$-dimensional Hamming cube gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on the $n$-dimensional Hamming cube which are determined by a weight which respects support of vectors (TS-metrics). We consider the known tilings of the Hamming cube and first determine which of them give rise to a perfect code. In the sequence, for those tilings that satisfy this condition, we determine all the TS-metrics that turns it into a perfect code. We also propose the construction of new perfect codes obtained by the concatenation of two smaller ones.

cs.IT↗

arXiv2019

Metrics which turn tilings into binary perfect codes

In this work, we consider tilings of the Hamming cube and look for metrics which turn the tilings into a perfect code. We consider the family of metrics which are determined by a weight and are compatible with the support of vectors (TS-metrics). We determine which of the tilings with small tiles or high rank can be a perfect code for some TS-metric and we characterize all such metrics. Finally, we show some procedures to obtain new perfect codes (relatively to TS-metrics) out of existing ones.

cs.IT↗

arXiv2019

Equivalence classes of small tilings of the Hamming cube

The study of tilings is a major problem in many mathematical instances, which is studied in two main different approaches: when considering the existence (or obstructions to the existence) of a tiling with a given tile and the other considering classification of tilings. Considering the Hamming cube $\mathbb{F}_2^n$, the small tilings, that is, tilings considering tiles with $8=2^3$ elements, were classified in \cite{vardy}. The authors list a total of $193$ different tiles. As the authors noted, many of those tiles can be obtained one from the other by a linear map. In this work, we are concerned with a particular class of linear maps, the class of permutations of coordinates. This is of interest since a permutation is an isometry of the Hamming cube, considering the Hamming metric. We show here that, up to an isometry, all those $193$ tiles can be reduced to $15$ classes. The proof is done by explicitly showing the permutation (represented in cycles) that identify each tile with a given representative.

cs.IT↗