The Classification of the Largest Caps in AG(5, 3)

Abstract

We prove that 45 is the size of the largest caps in AGð5; 3Þ; and such a 45-cap is always obtained from the 56-cap in PGð5; 3Þ by deleting an 11-hyperplane. # 2002 Elsevier Science (USA)

Sequences in abelian groups G of odd order without zero-sum subsequences of length exp(G)

“…By Theorem 2, we know that 42 is the size of the second largest complete cap in AG(5, 3). As a consequence and due to the uniqueness of the maximal affine 45-cap (see [6]) we have that, if we can construct a larger sequence with property D for all odd n, its restriction to n = 3 is contained in the affine part of the Hill cap.…”

“…For AG (5,3) it is known that the largest cap has size 45 [6]. The 42-cap obtained from the sequence here is complete, i.e.…”

Inverse zero-sum problems

“…, and in this connection it found a lot of attention by finite geometers (see [14,6] and the literature cited there).…”

Large caps

“…Vol. 76, 2003 Large caps 21 We sketch the construction in case q ≡ 3(mod 4). Use homogeneous coordinates (x 1 : x 2 : (4, q).…”

“…In fact, the Hill cap is the only 56-cap in PG(5, 3) [25]. It is a recent result from [21] that m 2 (AG(5, 3)) = 45 and the affine 45-cap is uniquely determined: it is contained in the Hill cap. There is hope to go one further step in the ternary case.…”