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  • 1.
    Nilson, Tomas
    et al.
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik.
    Heidtmann, Pia
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik.
    Inner balance of symmetric designs2014Inngår i: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 71, nr 2, s. 247-260Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.

  • 2.
    Nilson, Tomas
    et al.
    Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för ämnesdidaktik och matematik.
    Öhman, Lars-Daniel
    Umeå University, Department of Mathematics and Mathematical Statistics.
    Triple arrays and Youden squares2015Inngår i: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 75, nr 3, s. 429-451Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper addresses the question of when triple arrays can be constructed from Youden squares by removing a column together with the symbols therein, and then exchanging the role of columns and symbols. The scope of the investigation is limited to the standard case of triple arrays with {Mathematical expression}. For triple arrays with {Mathematical expression} it is proven that they can never be constructed in this way, and for triple arrays with {Mathematical expression} it is proven that there always exists a suitable Youden square and a suitable column for this construction. Further, it is proven that Youden square constructed from a certain family of difference sets never give rise to triple arrays in this way but always gives rise to double arrays. Finally, it is proven that all triple arrays in the single known infinite family, the Paley triple arrays, can all be constructed in this way for some suitable choice of Youden square and column.

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