The space mission called "Mariner 9" had gone to mars, took close photographs and were sent to the earth without errors. The photographs traveled millions of kilometers without contamination.
They have used what is called 'Hadamard code or Matrix'. Let us first see the construction of that matrix.
A = 1 1
1 0
In the above matrix, all the three entries are the same. The bottom right corner entry is just reversed. Hence, between the first and the second row, only one bit is different. We will extend the idea to 4 X 4 matrix.
B = 1 1 1 1
1 0 1 0
1 1 0 0
1 0 0 1
In the above matrix, the matrix A is repeated in all the three corners. Again, the matrix A is reversed in the right bottom corner. Here, between any two rows, two bits are different.
For example:
II row : 1010
IV row: 1001
Here last 2 bits are different. This is the property of Hadamard matrix.
Let each row in the matrix B represent a colour.
1111 - red
1010 - green
1100 -- blue
1001 -- black
A digital photograph is made of number of points or pixels (picture element). Each pixel is made of a colour or combination of colours. The colour info. in each pixel is coded using the above matrix by the Mariner (only an example) and sent to earth.
Suppose, we receive 1110, It is an error because no such row in the matrix. So error is successfully detected. It can corrected using the context of pixel in the photograph.
Hence the idea of Hadamard matrix is + +
+ -
Using this idea, we can build 8x8, or 16x16 Hadamard matrix. By posting the matrix B in 3 corners and reversing the B in the 4th corner, we can construct 8x8 matrix. In the higher order matrix, more bits will differ among the rows. Hence more errors can be detected and corrected efficiently. The price we have to pay for "the error free photo" is the lengthy code.
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