  Existing construction algorithms of block network-error correcting codes
require a rather large field size, which grows with the size of the network and
the number of sinks, and thereby can be prohibitive in large networks. In this
work, we give an algorithm which, starting from a given network-error
correcting code, can obtain another network code using a small field, with the
same error correcting capability as the original code. An algorithm for
designing network codes using small field sizes proposed recently by Ebrahimi
and Fragouli can be seen as a special case of our algorithm. The major step in
our algorithm is to find a least degree irreducible polynomial which is coprime
to another large degree polynomial. We utilize the algebraic properties of
finite fields to implement this step so that it becomes much faster than the
brute-force method. As a result the algorithm given by Ebrahimi and Fragouli is
also quickened.
