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 symbols from an alphabet of size 16. The value of maximum achievable entropy (in bits) is _______ .

A speech signal, band limited to 4 kHz , is sampled at 1.25 times the Nyquist rate. The speech samples, assumed to be statistically independent and uniformly distributed in the range -5 V to +5 V , are subsequently quantized in an $8-$ bit uniform quantizer and then transmitted over a voice - grade AWGN telephone channel. If the ratio of transmitted signal power to channel noise power is 26 dB , the minimum channel bandwidth required to ensure reliable transmission of the signal with arbitrarily small probability of transmission error (rounded off to two decimal places) is $\_\_\_\_$ kHz .

A binary random variable $X$ takes the value +2 or -2 . The probability $P(X=+2)=\alpha$. The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $X$ is maximum, is $\_\_\_\_$ .