I have the matrix H=[1 0 0 1 0 1 1;0 1 0 1 1 1 0;0 0 1 0 1 1 1] (3 rows,7 columns) which is the parity check matrix. And a codeword c is transmitted and r=[1 1 x y 1 1 1] is received. My job is to decode this. I guess i must multiply H with r' (transpose or r) so that will give me the syndrome s=[1+y;1+y;1+x].And i equate this s with the [0;0;0]. Then x=y=1. Is this right?I mean this is too easy to be true.I don't think that i am on the right track. PS:i guess this would be true if i had only one bit error...but i have 2 errors And from where do i know that H is systematic or nesystematic?
It's been many years since I've been through this, so take this with a grain of salt. 1) In general, an error correcting code that can correct all n-bit errors MAY be able to correct SOME errors greater than n-bits. 2) You are assuming that all of the bits other than x and y were received correctly. This is not the normal situation you are working with with applying an error correcting code. Instead, you are using this code as an erasure code. Erasure codes typically have greater strengths because they can make this assumption.