From 0850a1cd1c5c190ca5eb3af2cf1cb5e884587ee2 Mon Sep 17 00:00:00 2001 From: Florent Madelaine Date: Mon, 29 Jan 2024 11:20:32 +0100 Subject: [PATCH] =?UTF-8?q?double=20parit=C3=A9=20et=20targz=20MVAP=20en?= =?UTF-8?q?=20javascript?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 1InformationTheory.md | 33 +++++++++++++++++++++++++++++++++ 1InformationTheory.ods | Bin 20216 -> 24970 bytes MVaP/MVaP-1.0.tgz | Bin 0 -> 78200 bytes 3 files changed, 33 insertions(+) create mode 100644 MVaP/MVaP-1.0.tgz diff --git a/1InformationTheory.md b/1InformationTheory.md index d4caf4e..6fba6b5 100644 --- a/1InformationTheory.md +++ b/1InformationTheory.md @@ -36,3 +36,36 @@ At reception of some word $b'_1\ldots b'_nc'$ we check whether $f(b'_1\ldots b'_ This is used for low level transmission of information, in particular for ascii characters (since we tend to use powers of 2 when transmitting and storing information and we have one available bit when storing the 7 bits of the ascii encoding). [Details here](https://en.wikipedia.org/wiki/Parity_bit) + +## Correction + +We shall consider a last example which allows to detect and correct 1 error, and the algorithm to do this is quite simple. + +Assume that we have 9 bits of information. We write this bits in the form of a 3 times 3 matrix. We add 7 redundant bits, one at the end of each line, one at the bottom of each column, and one in the bottom right corner. Each redundant bit is the sum modulo 2 of the corresponding column / line. + +With this scheme we can detect 2 errors, but not correct them as there might be up to three codewords that are the nearest to a received message with 2 errors. + +We can always correct 1 error by recomputing the redundant data and compare it to the received data. +In particular, if the error is within the information part, the line / column of the bit to correct lies at the intersection of the line and colmun where the two redundant bits differ. + +We call this informally the matrix code below. + +## Hamming distance, Minimal distance of a code. + +Drawing on the board. + +Example for the repeating 3 times code. + +## Comparing codes. + +Examples with : +* Repetition code (1 bit of information, 2 redundant bits). Detects 2 errors, corrects 1. +* parity code (7 bits of information, 1 parity bit). Detects 1 error, corrects none. +* matrix code (9 bits of information, 7 redundant bits). Detects 2 errors, corrects 1. + +## Conclusion. + +Very quick discussion of linear code. + +Importance of encoding, detecting and correcting schemes that are efficient. + diff --git a/1InformationTheory.ods b/1InformationTheory.ods index aa825fddc9ef5a0ef547ba6d33271fdab3647568..ca14fcd87e908ed9b7a4ee8c52d743b5d9446bfb 100644 GIT binary patch literal 24970 zcmb4p1B|9kv*^r@y)!$uZQIzfZQHhO+kVHkZS2_Qj(3dr?)RT_ZgP`9|IO*7pH#ZL z)ZO(|r>YdB!N5^LK%hZDjFtE_LT$LB=s`e0{^@^*Kx{2-OIhogJJ^4NaUaOif)B{vVou$^74j`4Hik%ucWU3nHToIky+ZmbQlG zrp^q)mM*r24$l8AE;Kaszpm5Y_5ZiQ|AlM~?JUhqon7diOw4AIXY7I*zlYrLhMLZ< z2{J%HI8jb^SmpP#g=s)A+-ITe^u{HtDV6{(@|O;4#aVXanzOGAamS+@D+W|pjj6~c ze_FJh({Ojuca&~&Z>+V_5M<%qSp}EoQ+K-CQFH01h0<}r+sri;QF>%O@XB`+4=i&! z{)8WGj)R85j}^9jJo3TjnmvFqQ&)7kEJ*NtCgh9W=B_Mggf(nc>IrWpObir^g^`&I zaS6ull#?D|NqbP_8YsLv7&GYk63NRws0?8)g3MTf?-h5K4=NZJzg>KglO7GcY$X^Aurtof>>pVMk!VqJcZ9#hJnsJ@y=~fDerpRVk&!A(4t|t zN41y0BcQMTx$A>uO2GDIb>x&_*1Gb84#9=3*na`ThrG?eP=!Zd;4C!b;OCWDUEBh-KmWOZyU~s&MGz`N~K76&Us$9OXQkP 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