Please review/revisit the last post for end-of-term notifications and make sure everything is in place for you and you aren’t missing and deadlines or anything.
For the Hamming code example we used in the notes, i.e. with generating matrix $\begin{pmatrix} 1& 0 & 0 &0&1&1&0 \\ 0&1&0&0&1&0&1 \\ 0&0&1&0&0&1&1 \\ 0&0&0&1&1&1&1 \end{pmatrix}$:
Write out the parity check matrix and check that the first row of the generating matrix is a codeword.
Suppose you receive the transmission 1000101. Determine the syndrome.
Determine the most likely codeword that was sent.
In class, I listed out 8 elements of the example cyclic code given, which lives in the ring $R = \mathbb{F}_2[x]/(x^7-1)$.
I wrote out all the 8 codewords we found in class, i.e. $g(x)x^k$ for various $k$. Find this list.
Verify that $g(x)(x^3 + x^2+1) = x^7-1 = 0$. Explain why this means that $g(x)f(x) = g(x)f_1(x)$ for some $f_1(x)$ of degree $2$ or lower.
How many elements of the ring $R$ of degree $\le 2$ are there?
Explain why the above means the elements we already found are all the elements of this code.
Please let me know your requests for review videos! Just let me know what review question or general topic (bite-sized please), and I’ll make and post a short explanatory video. This is to make up for the last lecture and office hours I’ll be missing while travelling.