Abstract. All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in Q( √ −3) are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.In 1978 W. Feit [10] classified the unimodular hermitian lattices of dimensions up to 12 over the ring Z[ω] of Eisenstein integers, where ω is a primitive third root of unity. These lattices all have roots, that is, vectors of norm 2. In dimension 13, for the first time a unimodular lattice without roots appears [1,3]. In [2] the unimodular lattices in dimension 13 are completely classified. The root-free lattice turns out to be unique. It has minimal norm 3, and its automorphism group is isomorphic to the group Z 6 × PSp 6 (3) of order 2 10 .3 10 .5.7.13. The remaining lattices all have roots; the rank of the root system is 12 in all cases.In this paper, we classify the unimodular lattices in dimensions 14 and 15. There are exactly 58, respectively 259 classes of indecomposable lattices in these dimensions. Below, we list their root systems and the orders of their automorphism groups. Gram matrices for all lattices are available electronically via www.mathematik.uni-dortmund.de/~scharlauThere is only one root-free unimodular lattice of rank 14, and there are two root-free unimodular lattices of rank 15.The lattices without roots have minimal norm 3; they are extremal as introduced for unimodular Eisenstein lattices in [8], Chapter 10.7. They give rise to 3-modular extremal Z-lattices in twice the dimension, as defined by Quebbemann in [17]. See [8,19,20] for more information on extremal and modular lattices and their relation to modular forms. In this context, the lattices classified in this paper can be considered as complex structures on (extremal) 3-modular lattices. The question for existence, uniqueness, and possibly a full classification of extremal modular lattices has been an ongoing challenge, both computationally and theoretically, after the appearance of the influential paper [17].Let V be a vector space over Q( √ −3) with a positive definite hermitian product (, ). A lattice L in V is a finitely generated Z[ω]-module contained in V such that L contains a basis of V and (x, y) ∈ Z[ω] for all x, y ∈ L. More precisely, one

H. Cohn et al. proposed an association scheme of 64 points in R 14 which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal linesets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.

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In this paper we study those bent functions which are linear on elements of spreads, their connections with ovals and line ovals, and we give descriptions of their dual bent functions. In particular, we give a geometric characterization of Niho bent functions and of their duals, we give explicit formula for the dual bent function and present direct connections with ovals and line ovals. We also show that bent functions which are linear on elements of inequivalent spreads can be EA-equivalent.