# For Wednesday, Dec 6th, 2023:

1. For the Hamming code example we used in the notes, i.e. with generating matrix $\begin{pmatrix} 1& 0 & 0 &0&1&1&0 \\ 0&1&0&0&1&0&1 \\ 0&0&1&0&0&1&1 \\ 0&0&0&1&1&1&1 \end{pmatrix}: 1. Write out the parity check matrix and check that the first row of the generating matrix is a codeword. 2. Suppose you receive the transmission 1000101. Determine the syndrome. 3. Determine the most likely codeword that was sent. 2. In class, I listed out 8 elements of the example cyclic code given, which lives in the ring$R = \mathbb{F}_2[x]/(x^7-1)$. 1. Write out all the 8 codewords we found in class, i.e.$g(x)x^k$for various$k$. 2. I then pointed out that$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. 3. How many elements of the ring$R$of degree$\le 2$are there? 4. Explain why the above means the elements we already found are all the elements of this code. 3. Consider the cyclic code generated by$g(x) = x^3 + 3x^2 + x + 6$in the ring$R = \mathbb{F}_7[x]/(x^6-1)$. 1. Find a generating matrix. 2. What is the length of this code? 3. What dimension is this code? 4. What is the minimum distance of this code? 5. Divide$x^6-1$by$g(x)\$ (polynomial long division!)
6. We’ll continue this example in class.