MinT - Best Known (20−9, 20, s)-Nets in Base 2

Best Known (20−9, 20, s)-Nets in Base 2

(20−9, 20, 26)-Net over F₂ — Constructive and digital

Digital (11, 20, 26)-net over F₂, using

linear code embedding found in a computer search [i]

(20−9, 20, 48)-Net over F₂ — Upper bound on s (digital)

There is no digital (11, 20, 49)-net over F₂, because

1 times m-reduction [i] would yield digital (11, 19, 49)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹, 49, F₂, 8) (dual of [49, 30, 9]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(2¹⁸, 35, F₂, 8) (dual of [35, 17, 9]-code), but
      - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(2³⁰, 49, F₂, 14) (dual of [49, 19, 15]-code), but
      - discarding factors / shortening the dual code would yield linear OA(2³⁰, 48, F₂, 14) (dual of [48, 18, 15]-code), but
        adding a parity check bit [i] would yield linear OA(2³¹, 49, F₂, 15) (dual of [49, 18, 16]-code), but
        â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(20−9, 20, 54)-Net in Base 2 — Upper bound on s

There is no (11, 20, 55)-net in base 2, because

1 times m-reduction [i] would yield (11, 19, 55)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 550771 > 2¹⁹ [i]