Let \(q\) be a prime power. Then, as usual, \(A(q)\) is given by \[ \limsup_{g(K)\rightarrow\infty} N(K)/g(K), \] where \(K\) denotes a function field over \(\mathbb{F}_q\), \(N(K)\) is the number of places of degree 1 of \(K\), and \(g(K)\) is the genus of \(K\). J.-P. Serre [C. R. Acad. Sci., Paris, Sér. I 296, 397-402 (1983; Zbl 0538.14015)] used Hilbert class field towers to give a lower bound for \(A(q)\). Further results about \(A(q)\) using similar methods have been obtained by M. Perret [J. Number Theory 38, 300-322 (1991; Zbl 0741.11044)] and H. Niederreiter and C. Xing [Math. Nachr. 195, 171-186 (1998; Zbl 0920.11039)]. By splitting places of degree \(n\), the author is able to use Hilbert class field towers to obtain a lower bound for \(A(q^n)\) that is an improvement of Serre’s bound. In the final section, the author obtains lower bounds for \(A(3)\) and \(A(5)\) that improve bounds due to Niederreiter and Xing [op. cit.]. These improved bounds for \(A(3)\) and \(A(5)\) have also been obtained through different constructions by B. Anglès and C. Maire [Finite Fields Appl. 8, 207-215 (2002)].