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Let C be linear binary code of length n . For i € {1,2, ~n} let e(i) be the word with 1 in position i and 0 in all other positions. Suppose that H is a parity...

Question

Let C be linear binary code of length n . For i € {1,2, ~n} let e(i) be the word with 1 in position i and 0 in all other positions. Suppose that H is a parity check matrix for C. Let v, v € Zz. Theorem 14.6 says that C +v = C+v (that is, v and are in the same cosets) if and only if vHtr = U Htr Show that C is one-error correcting if and only if the columns of H are distinct and non-zero.

Let C be linear binary code of length n . For i € {1,2, ~n} let e(i) be the word with 1 in position i and 0 in all other positions. Suppose that H is a parity check matrix for C. Let v, v € Zz. Theorem 14.6 says that C +v = C+v (that is, v and are in the same cosets) if and only if vHtr = U Htr Show that C is one-error correcting if and only if the columns of H are distinct and non-zero.



Answers

Verify the uniqueness of $A$ in Theorem $10 .$ Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$
be a linear transformation such that $T(\mathbf{x})=B \mathbf{x}$ for some $m \times n$ matrix $B .$ Show that if $A$ is the standard matrix for $T,$ then $A=B .$ IHint: Show that $A$ and $B$ have the same columns.


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