double parité et targz MVAP en javascript

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.
+