On non-full-rank perfect codes over finite fields

Alexander M. Romanov

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arXiv · 1704.02627

On non-full-rank perfect codes over finite fields

Alexander M. Romanov

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Abstract

The paper deals with the perfect 1-error correcting codes over a finite field with $q$ elements (briefly $q$-ary 1-perfect codes). We show that the orthogonal code to the $q$-ary non-full-rank 1-perfect code of length $n = (q^{m}-1)/(q-1)$ is a $q$-ary constant-weight code with Hamming weight equals to $q^{m - 1}$ where $m$ is any natural number not less than two. We derive necessary and sufficient conditions for $q$-ary 1-perfect codes of non-full rank. We suggest a generalization of the concatenation construction to the $q$-ary case and construct the ternary 1-perfect codes of length 13 and rank 12.

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Alexander M. Romanov. 2017-04-09. On non-full-rank perfect codes over finite fields. https://arxiv.org/abs/1704.02627

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