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On full-rank perfect codes over finite fields. (Russian, English) Zbl 1374.94855
Diskretn. Anal. Issled. Oper. 23, No. 3, 107-123 (2016); translation in J. Appl. Ind. Math. 10, No. 3, 444-452 (2016).
Summary: We propose a construction of full-rank \(q\)-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by T. Etzion and A. Vardy [IEEE Trans. Inf. Theory 40, No. 3, 754–763 (1994; Zbl 0824.94029)]. 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 to the \(q\)-ary case. We propose a generalization of the notion of an \(i\)-component of a 1-perfect code and introduce the concept of an \((i, \sigma)\)-component of a \(q\)-ary 1-perfect code. 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 \(n\).
94B25 Combinatorial codes
94B60 Other types of codes
Full Text: DOI arXiv
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