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 $R = \mathbb{F}_2[x]/(x^7-1)$.

Write out all the 8 codewords we found in class, i.e. $g(x)x^k$ for various $k$.

I then pointed out that $g(x)(x^3 + x^2+1) = x^7-1$. 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.

Consider the cyclic code generated by $g(x) = x^3 + 3x^2 + x + 6$ in the ring $R = \mathbb{F}_7[x]/(x^6-1)$.

Find a generating matrix.

What is the length of this code?

What dimension is this code?

What is the minimum distance of this code?

Divide $x^6-1$ by $g(x)$ (polynomial long division!)