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arXiv · 1412.3795

Embedding in $q$-ary $1$-perfect codes and partitions

Denis S. Krotov·Evgeniya V. Sotnikova
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Abstract

We prove that every $1$-error-correcting code over a finite field can be embedded in a $1$-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting $1$-perfect code and can be obtained from it by repeated shortening. Further, we generalize the results to partitions: every partition of the Hamming space into $1$-error-correcting codes can be embedded in a partition of a space of some larger dimension into $1$-perfect codes. For the partitions, the embedding length is close to the theoretical bound for the general case and optimal for the binary case. Keywords: error-correcting code, $1$-perfect code, $1$-perfect partition, embedding

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Denis S. Krotov, Evgeniya V. Sotnikova. 2014-12-15. Embedding in $q$-ary $1$-perfect codes and partitions. https://doi.org/10.1016/j.disc.2015.04.014

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