Abstract
Improving Bruck's Completion-Theorem for nets, we show that a net of order k and degree k + 1 − δ can be extended to an affine plane, if 3k > 8δ3 − 18δ2 + 8δ + 4. As applications we obtain the following two theorems: A maximal partial t-spread in PG(2t + 1, q), q not a square, with deficiency δ > 0 satisfies 8δ3 − 18δ2 + 8δ + 4 ≥ 3q 2. There exists an absolute constant c such that every linear space with constant point degree n + 1 and minimum line degree n + 1 − a can be embedded in a protective plane of order n provided that n > ca 3.
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Communicated by A. Beutelspacher
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Metsch, K. Improvement of Bruck's completion theorem. Des Codes Crypt 1, 99–116 (1991). https://doi.org/10.1007/BF00157614
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DOI: https://doi.org/10.1007/BF00157614