Assuming that $x$ is a sequence of $l$ bits and $0 \le n < l$, let $R(x, n)$ denote the result of the left bitwise rotation of $x$ by $n$ bits. For example, if $x = 0100110001110000$, then $$\begin{array}{l}
R(x,0) = {\rm{0100110001110000}},\\
R(x,1) = {\rm{1001100011100000}},\\
R(x,2) = {\rm{0011000111000001}},\\
\ldots \\
R(x,15) = {\rm{0010011000111000}}.
\end{array}$$
Let $A \oplus B$ denote the result of the XOR operation for two sequences of $l$ bits. For example, $$0100110001110000 \oplus 1010010001000010 = 1110100000110010.$$
Let $H(x)$ denote the number of non-zero bits in $x$ (i.e. the Hamming weight of $x$).
Assuming that $x$ and $y$ are two bitstrings of the same length $l$, let $f(x, y)$ denote the minimal element (the smallest number) in the tuple $$\begin{array}{l}
(H(x \oplus y),\\
H(x \oplus R(y,1)),\\
H(x \oplus R(y,2)),\\
\ldots \\
H(x \oplus R(y,l - 1))).
\end{array}$$
Suppose that we have a TRNG which generates sequences of random bits. Generate a sequence of $L = k \times l$ bits. Split this sequence into $k$ words (so the length of each word is $l$): $w_0, w_1, \ldots, w_{k-1}$. Then compute the following tuple $T$ of numbers:
$$\begin{array}{l}
(f({w_0},{w_1}),\\
f({w_0},{w_2}),\\
\ldots \\
f({w_0},{w_{k - 1}}),\\
f({w_1},{w_2}),\\
f({w_1},{w_3}),\\
\ldots \\
f({w_1},{w_{k - 1}}),\\
f({w_2},{w_3}),\\
\ldots \\
f({w_{k - 2}},{w_{k - 1}})).
\end{array}$$
In other words, for any pair of words $(w_i, w_j)$ such that $i \neq j$, compute the corresponding $f({w_i},{w_j})$.
Question 1: given $k$ and $l$, how to compute the expected value of the minimal number $M_T$ in $T$?
Question 2: given $k$ and $l$, how to compute the expected value of the average number $A_T$ in $T$? Here the number $A_T$ is computed as follows: sum all elements of $T$, then divide the sum by the total number of elements in $T$.
The expected number here implies the number with the maximum probability. For example, the expected number of zero bits in a sequence of $l$ random bits is $l/2$.