Exact expressions for the maximal probability that all $k$-wise independent bits are 1

Let $M(n,k,p)$ denote the maximum probability of the event $X_1=X_2=\cdots =X_n=1$ under a $k$-wise independent distribution whose marginals are Bernoulli random variables with mean $p$. A long-standing question is to calculate $M(n,k,p)$ for all values of $n,k,p$. This question has been partially addressed by several authors, primarily with the goal of answering asymptotic questions. The present paper focuses on obtaining exact expressions for this probability. To this end, we provide closed-form formulas of $M(n,k,p)$ for $p$ near 0 as well as $p$ near 1.