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By N. S. Gopalakrishnan

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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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