Two Problems from Coding Theory

We suggest to prove the two following conjectures about the properties of some special functions. First consider the following polynome in $\lambda\in [0, (q-1)/q],\ q=2,3,\ldots $ $f^{q}_L (\lambda )=\sum\limits_{j_i :\ \sum_{i=1}^{q}j_i =L+1}\left( 1-\frac{\max\{ j_1 , \ldots ,j_q \}}{L+1}\right){L+1\choose j_1 ,\ldots ,j_{q}}\left(\frac{\lambda}{q-1}\right)^{L+1-j_q} (1-\lambda )^{j_q} .$ It arise in the problem of obtaining the upper bound for the rate of multiple pa cking of q–ary Hamming space. The problem is to prove that this function is ∩–convex.