Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.

Submanifolds of finite type were introduced by B.-Y. Chen about thirteen years ago [2], Many works have been done in characterizing or classifying submanifolds in Euclidean space with this notion. On the other hand, several authors studied submanifolds with finite type Gauss map. B.-Y, Chen and P. Piccinni studied compact submanifolds with finite type Gauss map [3]. And C. Baikoussis, B.-Y. Chen and L. Verstraelen classified ruled surfaces and tubes with finite-type Gauss map [1]. Recently Y. H. Kim studied surfaces in 3-dimensional Euclidean space £ 3 with 1-type Gauss map and he proved that the only co-closed surfaces in E 3 with 1-type Gauss map are spheres and circular cylinders [6]. In this paper we study surfaces in E z with 1-type Gauss map without the assumption of co-closedness and obtain the following theorem. THEOREM. Let M be an orientable, connected surface in E B . Then M has l-type Gauss map if and only if M is an open part of a sphere or an open part of a circular cylinder.

We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is one-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield some new information about the smoothness of the pseudoriemannian conjugate locus. We also obtain the multiplicities of all conjugate points. (2000): primary 53C50; secondary 22E25, 53B30, 53C30. Mathematics Subject Classifications

Abstract. Let n be a 2-step nilpotent Lie algebra which has an inner product ⟨ , ⟩ and has an orthogonal decomposition n = z ⊕ v for its center z and the orthogonal complement v of z. Then Each element z of z defines a skew symmetric linear map Jz : v −→ v given by ⟨Jzx, y⟩ = ⟨z, [x, y]⟩ for all x, y ∈ v. In this paper we characterize Jacobi fields and calculate all conjugate points of a simply connected 2-step nilpotent Lie group N with its Lie algebra n satisfying J 2 z = ⟨Sz, z⟩A for all z ∈ z, where S is a positive definite symmetric operator on z and A is a negative definite symmetric operator on v.