Classification Algorithms for Codes and Designs by Petteri Kaski

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 structures of order 15]. this technique was once offered, and its tedious and di?cult execution und- taken, by means of Mr. Cole. F. N. Cole, L. D. Cummings, and H. S. White (1917) [129] The heritage of classifying combinatorial gadgets is as previous because the background of the gadgets themselves. within the mid-19th century, Kirkman, Steiner, and others turned the fathers of recent combinatorics, and their paintings – on a variety of gadgets, together with (what grew to become later referred to as) Steiner triple structures – resulted in a number of classi?cation effects. virtually a century prior, in 1782, Euler [180] released a few effects on classifying small Latin squares, yet for the ?rst few steps during this course one may still truly move not less than way back to historic Greece and the facts that there are precisely ?ve Platonic solids. probably the most impressive achievements within the early, pre-computer period is the classi?cation of the Steiner triple platforms of order 15, quoted above. An exhausting job that, this day, no good individual might try via hand calcu- tion. simply because, aside from occasional parameters for which com- natorial arguments are e?ective (often to end up nonexistence or uniqueness), classi?cation more often than not is set algorithms and computation.

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113. 105 form an OA(3, 3). 112, this can be transformed into a Latin square of side 3. 0 1 2 1 2 0 2 0 1 There are several definitions concerning indistinguishable Latin squares. 112 for conveniently introducing these concepts. 114. Two Latin squares are said to be in the same 1. main class if the corresponding orthogonal arrays are isomorphic, 2. type if the corresponding orthogonal arrays can be mapped onto each other using a permutation of the first two rows, value permutations in the rows, and a permutation of the columns, 3.

A Latin square of side n can be transformed into an orthogonal array OA(3, n) and vice versa. 40 2 Graphs, Designs, and Codes Proof. From a Latin square of side n, construct on OA(3, n) with columns T (i, j, L(i, j)) for i, j ∈ Zn . To construct a Latin square from an orthogonal array, proceed in the opposite direction. 113. 105 form an OA(3, 3). 112, this can be transformed into a Latin square of side 3. 0 1 2 1 2 0 2 0 1 There are several definitions concerning indistinguishable Latin squares.

Among several reasonable functions and definitions of optimality, we here study dmax (n, k) = max{d : there exists an [n, k, d]2 code} and call an [n, k, dmax (n, k)]2 code optimal . Similar functions may be introduced for q > 2. We will next discuss some basic properties of linear codes; see [388] or any introductory textbook for proofs of these results. Two types of matrices are closely related to linear codes. 91. A k ×n matrix G is a generator matrix for an [n, k]q linear code C if uG generates the codewords of C for u ∈ Fkq .

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