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oa Litnet Akademies : 'n Joernaal vir die Geesteswetenskappe, Natuurwetenskappe, Regte en Godsdienswetenskappe - Enumerasie van self-ortogonale Latynse vierkante van orde 10 : afdeling : natuurwetenskappe

 

Abstract

'n Latynse vierkant van orde n is 'n n x n-skikking van die simbole Z = {0, 1, 2, . . . n - 1} sodat elke ry en elke kolom van die skikking elke element van Z bevat. Indien die inskrywing in ry i ∈ Z en kolom j ∈ Z van 'n Latynse vierkant L aangedui word deur L(i; j), is twee Latynse vierkante L en L' ortogonaal as die geordende pare (L(i; j); L'(i; j)) almal verskillend is soos wat i en j oor Z varieer. 'n Latynse vierkant is verder self-ortogonaal indien die vierkant en sy transponent ortogonaal is, en idempotent indien L(i; i) = i vir alle i ∈ Z. Twee self-ortogonale Latynse vierkante L en L' is in dieselfde (ry, kolom)-paratoopklas indien daar twee permutasies p en q bestaan sodat as p op die rye en kolomme van L en q op die simbole van L toegepas word, die vierkant L of op L' of op die transponent van L' afgebeeld word. Verder is L en L' in dieselfde isomorfismeklas indien een permutasie toegepas op die rye, kolomme en simbole van L, die vierkant L afbeeld op L', en in dieselfde transponent-isomorfismeklas indien so 'n permutasie L op L' of op sy transponent afbeeld. Die groottes van bogenoemde ekwivalensieklasse van self-ortogonale Latynse vierkante, asook die getal idempotente self-ortogonale Latynse vierkante van orde hoogstens 9, is in die literatuur gedokumenteer. In hierdie artikel word hierdie ekwivalensieklasse van selfortogonale Latynse vierkante van orde 10 in 'n groot parallelle berekeningspoging getel.

A Latin square of order n is an n x n array of the symbols Z = {0, 1, 2, . . . n - 1} such that every row and every column of the array contains each element of Z. Denote the entry in row i ∈ Z and column j ∈ Z of a Latin square L by L(i; j). Two Latin squares L and L' are orthogonal if the ordered pairs (L(i; j); L0(i; j)) are all distinct as i and j vary over Z. Furthermore, a Latin square is self-orthogonal if the square is orthogonal to its transpose, and idempotent if L(i; i) = i for all i ∈ Z. Two self-orthogonal Latin squares L and L' are in the same (row, column)-paratopism class if there exist two permutations p and q such that, if p is applied to the rows and columns of L and q to the symbols of L, then L is mapped to L' or to the transpose of L'. Moreover, L and L' are in the same isomorphism class if one permutation applied to the rows, columns and symbols of L maps it to L', and in the same transpose isomorphism class if such a permutation maps L to L' or its transpose. The sizes of the above equivalence classes of self-orthogonal Latin squares, as well as the number of idempotent self-orthogonal Latin squares of order at most 9, have been documented in the literature. The equivalence classes of self-orthogonal Latin squares of order 10 are enumerated in this paper by means of a large parallel computing effort.

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/content/litnet/7/3/EJC62277
2010-12-01
2016-12-03
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