Chapter Questions
Prove the following about degrees of Boolean polynomials:(1) For nonzero Boolean polynomials $f$ and $g$, the degree of $f g$ is the sum of the degrees of $f$ and $g$.(2) The degree of $f+g$ is the maximum of the degrees of $f$ and $g$.
Find a binary $(15,5)$-code correcting triple errors. (Hint: use a punctured Reed-Muller code.)
When using $\mathcal{R}(1,3)$, decode 01111100 . Verify the correctness of your decoding. Encode information bits 1011.
When using $\mathcal{R}(2,3)$, decode 0111100 .
When using $\mathcal{R}(2,4)$, decode 1111111011111111 .
What Boolean polynomial has the truth table(1) 10100110 ?(2) 1010011010100110 ?
Find the truth table of the Boolean polynomial $1+x_0+x_1 x_2$(1) as a function of three variables,(2) as a function of four variables.
What is the relationship between simplex codes (8.2) and the ReedMuller codes $\mathcal{R}(1, m)$ ?
Compute the number of 2-flats in the Euclidean geometry in $\mathbf{Z}_2^m$.
Prove that each hyperplane $L$ in the Euclidean geometry in $\mathbf{Z}_2^m$ has the property that its complement $\mathbf{Z}_2^m-L$ is a hyperplane too.
Is every Boolean function a characteristic function of some flat? Characterize such functions! (Hint: express each flat as an intersection of hyperplanes.)