A decomposition of the 2-design formed by the planes in AG(2n,3)

It is well known that for a prime power s and a positive integer m, the set of d-flats in AG(m,s) forms a 2-design. In this article, it is shown that the 2-design formed by the 2-flats in AG(m,3) for even m can be decomposed into more subdesigns than a previously known decomposition. Exact calculation of the number of the resulting subdesigns is also demonstrated by examining the distribution of points in cyclotomic cosets.