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This can be a description of the underlying ideas of algebraic geometry, a few of its very important advancements within the 20th century, and a few of the issues that occupy its practitioners this day. it's meant for the operating or the aspiring mathematician who's unexpected with algebraic geometry yet needs to realize an appreciation of its foundations and its ambitions with no less than necessities. Few algebraic must haves are presumed past a easy direction in linear algebra.

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Xn (mod 2k) ∈ C Λ(C) = √ 2k is an n-dimensional unimodular lattice with the minimum norm min{ dE2k(C) , 2k}. Moreover, if C is Type II, then Λ(C) is Type II. Proof Clearly Λ(C) is an n-dimensional lattice. Let a1 , a2 ∈ Λ(C). Then ai = √1 (ci + 2kzi ), where ci ∈ C and zi ∈ Zn for i = 1, 2. Then [a1 , a2 ] = 1 ([c1 , c2 ] + 2k 2k 2k[c1 , z2 ] + 2k[c2 , z1 ] + 4k 2 [z1 , z2 ]) ∈ Zn since [c1 , c2 ] is a multiple of 2k. Thus, Λ(C) is integral. √Note that 2kZn ⊂ √ 2kΛ(C) ⊂ Zn . It is (2kZn ) = (2k)n and [ 2kΛ(C) : 2kZn ] = √easy to see that Vn/2 n/2 =√1 = det Λ, that is, (2k) .

2 for more details. 6 More Problems Related to a Prize Problem In this section, we further describe one of the long-standing open problems in algebraic coding theory. This is about the existence of a binary self-dual [72, 36, 16] code. We refer to [Ki1]. Let C be a binary Type I code, and C0 the doubly even subcode C0 of C (that is, the subcode of C consisting of all codewords of weight ≡ 0 (mod 4)). 6 More Problems Related to a Prize Problem 45 Sloane defined the shadow S of C by S := C0⊥ \C [CS3].

Bn ) ∈ GF(q)n be regarded as a block, and p = (p1 , . . , pn ) ∈ GF(q)n be regarded as a point. We say that b covers p, or p is in b, provided that (a) supp(p) ⊂ supp(b) and (b) for all i ∈ supp(b), (at least) one of the following conditions holds: (i) pi = bi , (ii) pi = 0. More generally, a q-ary t-(v, k, λ) design D = (P , B) is a pair consisting of the set P ⊂ GF(q)n of elements (called points) of weight t and a collection B of weight k elements of GF(q)n (called blocks) such that every point p ∈ P is covered by exactly λ blocks.

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