Let C be a binary code with parity check matrix

H $=$

$$

$$

$$

$$

$$

$$

$000000001111111111111111$

$111111110000000011111111$

$000011110000111100001111$

$001100110011001100110011$

$010101010101010101010101$

$$

$$

$$

$$

$$

$$

That is$,$ the columns of H are all possible binary vectors of length $5$ that do not start

with two zeros.

$($a$)$ What is the dimension of C$?$

$($b$)$ Let A be a standard array for C$.$ As discussed in class, the standard array is

constructed so that the first column consists of coset leaders for that row $($not

necessarily unique$).$ How many entries in the first column of the standard array

have weight $1?$