Ex q=2, k=2, n=
One way to define a code is through the parity check matrix. Ex Parity codes have H= (there are k-ones)
then G=( P)
Ex q=1, k=3, n=7
Ex q=4, k=2, n=5, dim(c)=n-k=5-2=3,
H=( ), now find peicewise linearly independent vectors in
= 0, 1, a, b (a=x, b=x+1)
Where was the error in ?
*Can we change one element to get a codeword?
Theorem Hamming codes have a minimum distance of 3. (Perfect) we can fix one error
If x had been or some such vector, then there would have only been one error and we could have fixed it. Below is a second applet displaying how a code works. This applet encodes words using the Hamming Code( ). This is the same code that was used in examples in this section. At the top, enter a word to be encoded. Upon hitting the ``encode" button, a box displaying the binary representation of the word will appear. Below the box with the binary string are boxes that show the encoded words over . Because this code takes in words of four numbers, and our letters are represented by six numbers, the encoded blocks do not correspond to letters individually. After the codewords appear feel free to change the numbers around, simulating how errors are created. Press the ``parity check" button and the number of the first block with an error will appear. This applet only takes words six letters in length. If the word entered is too short, it will be padded with spaces. If the word is too long then the word will be trimmed to the appropriate length.