The generator matrix of a $(6,3)$ binary linear block code is given by
$$ G=\left[\begin{array}{llllll} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{array}\right] $$
The minimum Hamming distance $d_{\min }$ between codewords equals___________ (answer in integer).
A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random variable independent of $S$. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $S$.
Suppose the probability of symbol estimation error $P_e$ is expressed as follows:
$$ P_e=\alpha P(N>1), $$
where $P(N>1)$ denotes the probability that $N$ exceeds 1 .
What is the value of $\alpha$ ?
Consider a real-valued random process
$$ f(t)=\sum\limits_{n=1}^N a_n p(t-n T), $$
where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and 0 otherwise. The coefficients $a_n$ are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the correct option.
(i) The mean of the process $f(t)$ is independent of time $t$.
(ii) The autocorrelation function $E[f(t) f(t+\tau)]$ is independent of time $t$ for all $\tau$. (Here, $E[\cdot]$ is the expectation operation.)
The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)$ and $\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<\frac{1}{2}$.
Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?