By Petteri Kaski

A new starting-point and a brand new strategy are considered necessary, to insure an entire [classi?cation of the Steiner triple platforms of order 15]. this technique was once provided, and its tedious and di?cult execution und- taken, by way of Mr. Cole. F. N. Cole, L. D. Cummings, and H. S. White (1917) [129] The heritage of classifying combinatorial gadgets is as outdated because the heritage of the gadgets themselves. within the mid-19th century, Kirkman, Steiner, and others turned the fathers of contemporary combinatorics, and their paintings – on a variety of items, together with (what grew to become later often called) Steiner triple platforms – resulted in numerous classi?cation effects. nearly a century previous, in 1782, Euler [180] released a few effects on classifying small Latin squares, yet for the ?rst few steps during this course one should still truly cross no less than way back to old Greece and the facts that there are precisely ?ve Platonic solids. probably the most striking achievements within the early, pre-computer period is the classi?cation of the Steiner triple platforms of order 15, quoted above. An laborious activity that, at the present time, no good individual might try by way of 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 quite often is ready algorithms and computation.

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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 . The unique solution n−1 (up to permutation of the indices) to i=0 ai = A with |ai − aj | ≤ 1 for all i, j is ai = (A + i)/n . 82 (Generalized Plotkin bound).

Cn ) and c = (c1 , c2 , . . , cn ), is c · c = c1 c1 + c2 c2 + · · · + cn cn , with addition and multiplication carried out in Fq . There are also other possible inner products, such as the Hermitian inner product, which is often preferred for F4 [497]. 98. Given a linear code C, its dual code is deﬁned as C ⊥ = {c : c · c = 0 for all c ∈ C}. A linear code C is called self-orthogonal if C ⊆ C ⊥ and self-dual if C = C ⊥ . Self-dual and self-orthogonal codes are surveyed in [497]. Using basic linear algebra, one may prove that the dual code is necessarily linear and that for codes of length n, dim(C ⊥ ) = n − dim(C).

Given one of the following objects, it can be transformed into the others: 1. a complete set of MOLS of side n, 2. an OA(n + 1, n), 3. an (n + 1, n2 , n)n OE code, 4. an aﬃne plane of order n, 5. a projective plane of order n. Proof. 116. (2) → (3): By the deﬁnition of an OA(n+1, n), two columns can agree in at most one row; otherwise there would exist a 2×n2 subarray that would contain 42 2 Graphs, Designs, and Codes a pair of values more than once. Hence, the columns of such an array can be viewed as codewords of a code with minimum distance n, an (n + 1, n2 , n)n code.