By Edmund Runggaldier
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1), a nonnull codeword in C is equivalent to a deﬁning pencil of d(B). The mathematical connection between sn−k designs and algebraic codes was established by Bose (1961). For a codeword (or vector) u = (u1 , . . , un ), the Hamming weight wt(u) is the number of its nonzero components. For two codewords u = (u1 , . . , un ) and w = (w1 , . . , wn ), the Hamming distance dist(u, w) = wt(u − w) is the number of j’s with wj = uj . The minimum distance of a code C is the smallest Hamming distance between any two distinct codewords of C.
1. Let d(B1 ) and d(B2 ) be two sn−k designs. Let r be the smallest integer such that Ar (B1 ) = Ar (B2 ). Then d(B1 ) is said to have less aberration than d(B2 ) if Ar (B1 ) < Ar (B2 ). A design is called a minimum aberration (MA) design if no other design has less aberration than it. Clearly, the resolution of a design d(B) equals the smallest integer j suchthat Aj (B) > 0. Hence in any given context, an MA design has the highest possible resolution as well. 2), the wordlength patterns of d(B1 ) and d(B2 ) are given by (0, 0, 1, 3, 0) and (0, 0, 2, 1, 1) respectively.
There are n−k n−k + + · · · = 2n−k−1 1 3 such points. , no three of these are linearly dependent. Thus there exists a collection of 2n−k−1 points of P G(n − k − 1, 2) such that no three of these points are linearly dependent. 2. 2. Let s = 2, n = 8, k = 4. 3) holds. The points of P G(3, 2) having an odd number of 1’s are (1, 0, 0, 0) , (0, 1, 0, 0) , (0, 0, 1, 0) , (0, 0, 0, 1) , (1, 1, 1, 0) , (1, 1, 0, 1) , (1, 0, 1, 1) , (0, 1, 1, 1) . Let T be the set of these eight points, no three of which are linearly dependent.