Homework SourseIf G IkA is a generator matrix for the n kcodec in standard
If G = [I_k|A] is a generator matrix for the [n, k]codec in standard form, then H = [-A^T|I_n - k] is a parity check matrix for C. Proof: We clearly have HG^T = -A^T + A^T = O. Thus C is contained in the kernel of the linear transformation x rightarrow Hx^T. As rank n - k, this linear transformation has kernel of dimension k, which is also the dimension of C. The result follows. Prior to the statement of Theorem 1.2.1, it was noted that the rows of the (n -k) times n parity check matrix satisfying (1.1) are independent. Why is that so?![If G = [I_k|A] is a generator matrix for the [n, k]codec in standard form, then H = [-A^T|I_n - k] is a parity check matrix for C. Proof: We clearly have HG^T](/WebImages/47/if-g-ika-is-a-generator-matrix-for-the-n-kcodec-in-standard-1148713-1761618064-0.webp "If G = [I_k|A] is a generator matrix for the [n, k]codec in standard form, then H = [-A^T|I_n - k] is a parity check matrix for C. Proof: We clearly have HG^T")
Solution
For G=[P|Ik],define the matrix H=[In-k |PT]
The size of H is (n-k)xn
It follows that GHT=0
Since c=mG, then cHt=mGHT-0
H=1 0 0 1 0 1 1
0 1 0 1 1 1 0
0 0 1 0 1 1 1
No column of H can be all zeros,or else an error in the corresponding code vector position would not affect the syndrome and would be undetectable.
All columns of H must be unique.If two columns are Identical errors corresponding to these code word
Locations will be indistinguishable.
A matrix H is called a parity-check matrix for a linear code C if the columns of H form a basis for the dual code Ci.
A matrix H is a parity-check matrix for some linear code iff the columns of H are linearly independent.
