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- W2012485718 abstract "In this paper, we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given Hamming code <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$C$</tex></formula> of length <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$n=(q^m-1)/(q-1)$</tex> </formula> over <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>${BBF}_q$</tex> </formula> with a parity check matrix <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$H_m$</tex> </formula> , we define a new linear code <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$C_{(m,r)}$</tex></formula> of length <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex Notation=TeX>$n$</tex></formula> over <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex Notation=TeX>${BBF}_{q^r},$</tex></formula> <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex Notation=TeX>$r geq 2$</tex></formula> , with this parity check matrix <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$H_m$</tex></formula> . The resulting code <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$C_{(m,r)}$</tex></formula> is completely regular with covering radius <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$rho = min{r,m}$</tex></formula> . We compute the intersection numbers of such codes and we prove that Hamming codes are the only codes that, after lifting the ground field, result in completely regular codes. Finally, we also prove that extended perfect (Hamming) codes, for the case when extension increases their minimum distance, are the only codes that, after lifting the ground field, result in uniformly packed (in the wide sense) codes." @default.
- W2012485718 created "2016-06-24" @default.
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- W2012485718 date "2011-09-01" @default.
- W2012485718 modified "2023-10-17" @default.
- W2012485718 title "On Lifting Perfect Codes" @default.
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- W2012485718 doi "https://doi.org/10.1109/tit.2010.2103410" @default.
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