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 system 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 background of classifying combinatorial gadgets 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 quite a few items, together with (what grew to become later referred to as) Steiner triple platforms – resulted in a number of classi?cation effects. virtually a century past, in 1782, Euler [180] released a few effects on classifying small Latin squares, yet for the ?rst few steps during this path one should still really cross not less than way back to historical Greece and the facts that there are precisely ?ve Platonic solids. 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 exhausting activity that, this day, no good individual could try by means 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 in most cases is set algorithms and computation.

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Example text

Lk (i, j)) for i, j ∈ Zn . In the opposite direction, k MOLS are obtained from an OA(k + 2, n). 116 is needed in the following definition. 117. Two sets of MOLS are isomorphic if the corresponding orthogonal arrays are isomorphic. 118. 105 can be transformed into two MOLS of side 3. 0 1 2 1 2 0 2 0 1 0 1 2 2 0 1 1 2 0 How many MOLS of side n can there be (more precisely, what is the maximum k, such that an OA(k, n) exists)? An upper bound on this number is given by the following theorem. 119.

There are a few central parameters related to codes. 74. The minimum (Hamming) distance of a code C with at least two codewords is d(C) = min{dH (x, y) : x, y ∈ C, x = y}. 75. The covering radius of a code C is R(C) = max{dH (x, C) : x arbitrary}, where dH (x, C) = min{dH (x, c) : c ∈ C}. The minimum distance d(C) of a code C is the most important parameter in the study of error-correcting and error-detecting codes because it measures the ability of the code to sustain transmission errors. Namely, if only words of C are used in the transmission of information, then up to d(C) − 1 bit errors in a word can be detected and up to (d(C) − 1)/2 bit errors can be corrected (by correcting a received word to the closest codeword).

73. A code C ⊆ Fnq is linear if it forms a subspace of the vector space Fnq . Otherwise C is nonlinear. Unless mentioned otherwise, the codes we study in this work are unrestricted , that is, either nonlinear or linear. Linear codes are also considered in their own right; an introduction to these is given in Sect. 3. Whenever unrestricted codes are considered, we assume without loss of generality that C ⊆ Zqn . There are a few central parameters related to codes. 74. The minimum (Hamming) distance of a code C with at least two codewords is d(C) = min{dH (x, y) : x, y ∈ C, x = y}.

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