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Enumerasie van self-ortogonale Latynse vierkante van orde 10

 

 Opsomming 

’n Latynse vierkant van orde n is ’n n×n-skikking van die simbole Zn = {0, 1, 2, ..., n-1}sodat elke ry en elke kolom van die skikking elke element van Zn bevat. Indien die inskrywing in ry iZn en kolom jZn 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 Zn 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 iZn.


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 óf op L′ óf 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 só ’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 self-ortogonale Latynse vierkante van orde 10 in ’n groot parallelle berekeningspoging getel.

 

Abstract

A Latin square of order n is an n×n array of the symbols Zn = {0, 1, 2, ..., n-1} such that every row and every column of the array contains each element of Zn. Denote the entry in row iZn and column j Zn of a Latin square L by L(i, j). Two Latin squares L and L′ are orthogonal if the ordered pairs (L(i, j), L′(i, j)) are all distinct as i and j vary over Zn. Furthermore, a Latin square is self-orthogonal if the square is orthogonal to its transpose, and idempotent if L(i, i) = i for all iZn.


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.

 

Klik hier om die volledige artikel in PDF-formaat af te laai.

 

Click here to read the extended abstract in English.

 



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