Coding theory: Find the right code word

[fiksx]

Homework Statement

Error correction can be performed on 1010101 after reception, i need to find the right code <br>

Relevant Equations

i know that the polynomial for the received code is $x^6+x^4+x2+1$

Error correction can be performed on 1010101 after reception, i need to find the right code <br>
i know that the polynomial for the received code is $$x^6+x^4+x2+1$$
when i try to find the error pattern,by long division, $$r(x)/g(x)$$
the remainder is $$z^2+z^2+1$$ xor $$z^2+z+1$$ so the remainder is $$z^2+z+1$$ or $$z^2+z$$?
and also how can i know the coset leader? do i need to divide $$e^i$$ for i=1 until 6 and find the one that have same remainder with the error?

 

[rcgldr]

I'm assuming this is a single bit correcting BCH code, for a codeword up to 7 bits (4 data, 3 ecc).
The field is GF(2^3), which could be g(x) = x^3 + x^2 + 1 or g(x) = x^3 + x + 1.
If g(x) = x^3 + x^2 + 1, r(x) mod g(x) = x^2, which doesn't match the question.
If g(x) = x^3 + x + 1, r(x) mod g(x) = x^2 + x, which matches the question.
I'm assuming g(x) = x^3 + x + 1, with α = x + 0
Syndrome(1) = r(α ) mod g(x) = r(x) mod g(x) = x^2 + x.
The error location is i , where α^i mod g(x) = x^2 + x.
The antilog table is α^{0,1,2,3,4,5,6} mod g(x) = {1, x, x^2, x+1, x^2+x, x^2+x+1, x^2+1}.
In this case i = 4.
So bit 4 (the 5th bit from the right) is in error, and the corrected code word is 1000101.

A codeword with all zero bits except bit 4 produces the same remainder:
x^4 mod g(x) = x^2 + x ... or 0010000 mod 1011 = 110.
Codewords {0000001, 0000010, ... , 1000000} mod 1011 follow the same pattern as the antilog table.