Abstract
In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q 2 − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q 2 − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.
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Communicated by J. D. Key.
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Landjev, I., Storme, L. A study of (x(q + 1), x; 2, q)-minihypers. Des. Codes Cryptogr. 54, 135–147 (2010). https://doi.org/10.1007/s10623-009-9314-y
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DOI: https://doi.org/10.1007/s10623-009-9314-y