Hamming(7, 4) has linear encoding, but not linear decoding. For example: 0000 encodes to 0000000 which means (with the error-correcting property) that: 0000001 decodes to 0000 0000010 decodes to 0000 0000100 decodes to 0000 0001000 decodes to 0000 0010000 decodes to 0000 0100000 decodes to 0000 1000000 decodes to 0000 If the decoding function were linear, it would therefore be identically zero (which it isn't). Best wishes, Adam P. Goucher
Sent: Wednesday, January 16, 2019 at 6:14 PM From: "Mike Stay" <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] *linear* decoding of ECC code ?
On Wed, Jan 16, 2019 at 10:51 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
To summarise, there is no code which satisfies each of:
(i) linear decoding; (ii) correction of arbitrary single-bit errors; (iii) ability to actually store information.
Hamming(7,4) is linear, corrects arbitrary single bit errors, and stores four bits of information. Is there some extra implicit constraint I'm missing?
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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