Geodesic
On full-rank perfect codes over finite fields
Diskretnyj analiz i issledovanie operacij, Tome 23 (2016) no. 3, pp. 107-123.

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We propose a construction of full-rank $q$-ary $1$-perfect codes over finite fields. This is a generalization of the construction of full-rank binary $1$-perfect codes by Etzion and Vardy (1994). The properties of the $i$-components of q-ary Hamming codes are investigated and the construction of full-rank $q$-ary $1$-perfect codes is based on these properties. The switching construction of $1$-perfect codes is generalized for the $q$-ary case. We propose a generalization of the notion of $i$-component of a $1$-perfect code and introduce the concept of an $(i,\sigma)$-component of $q$-ary $1$-perfect codes. We also present a generalization of the Lindström–Schönheim construction of $q$-ary $1$-perfect codes and provide a lower bound for the number of pairwise distinct $q$-ary $1$-perfect codes of length $n$. Bibliogr. 16.
Keywords: Hamming code, nonlinear perfect code, full-rank code, $i$-component. 
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A. M. Romanov. On full-rank perfect codes over finite fields. Diskretnyj analiz i issledovanie operacij, Tome 23 (2016) no. 3, pp. 107-123. http://geodesic.mathdoc.fr/item/DA_2016_23_3_a6/

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