Classification Algorithms for Codes and Designs - download pdf or read online
By Petteri Kaski
A new starting-point and a brand new strategy are considered necessary, to insure a whole [classi?cation of the Steiner triple platforms of order 15]. this technique was once provided, and its tedious and di?cult execution und- taken, through Mr. Cole. F. N. Cole, L. D. Cummings, and H. S. White (1917)  The background of classifying combinatorial items is as previous because the background of the items themselves. within the mid-19th century, Kirkman, Steiner, and others turned the fathers of contemporary combinatorics, and their paintings – on a number of items, together with (what turned later referred to as) Steiner triple structures – ended in numerous classi?cation effects. nearly a century past, in 1782, Euler  released a few effects on classifying small Latin squares, yet for the ?rst few steps during this course one may still truly cross at the very least way back to historical Greece and the facts that there are precisely ?ve Platonic solids. some of the most awesome achievements within the early, pre-computer period is the classi?cation of the Steiner triple platforms of order 15, quoted above. An arduous activity that, this day, no good individual could try via hand calcu- tion. simply because, apart from occasional parameters for which com- natorial arguments are e?ective (often to turn out nonexistence or uniqueness), classi?cation often is set algorithms and computation.
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Extra resources for Classification Algorithms for Codes and Designs
82. The Plotkin bound is of central importance in the study of these codes. This bound was proved by Plotkin  for the binary case. The q-ary generalization to be presented here is from . 81. Let i=0 ai = A, where the variables ai are nonnegative inn−1 tegers. Then i=0 a2i attains its smallest value when ai = (A + i)/n . Proof. We ﬁrst show that the smallest value cannot be attained if there exist i and j such that ai − aj ≥ 2. Namely, then we can substitute the values ai and aj by ai − 1 and aj + 1, respectively, to get an even smaller value, since (ai − 1)2 + (aj + 1)2 = a2i + a2j + 2(aj − ai + 1) < a2i + a2j .
16) M≤ 2 w − wn + δn provided that the denominator is positive. 16), then the codewords of any such code deﬁne the rows of an incidence matrix of a 2-(v, k, λ) design with v = M , k = wM/n, λ = w − δ, b = n, and r = w; and vice versa. 87 (see also the subsequent remark), the second Johnson bound can be used to prove that in constructing a BIBD point by point along the lines of Sect. 1, one missing row can always be completed (in a unique way). A similar result – proved with a diﬀerent technique – holds in constructing Steiner systems block by block [1, Theorem 13] (such methods are considered in Sects.
110. There exists an OAλ (2t, k, 2) if and only if there exists an OAλ (2t + 1, k + 1, 2). Proof. 109. Given an OAλ (2t, k, 2), we form an OAλ (2t + 1, k + 1, 2) by adding a new row of 0s, and juxtaposing (that is, putting side by side) this array and its complement (obtained by replacing the 0s by 1s and the 1s by 0s). To check the parameters of the new array, we have to look at sets of 2t + 1 rows. Obviously, if the new row is among the 2t + 1 rows, the result is clear, so it suﬃces to consider sets of 2t + 1 rows that do not contain the new row.
Classification Algorithms for Codes and Designs by Petteri Kaski