MinT - Best Known (66−33, 66, s)-Nets in Base 2

Best Known (66−33, 66, s)-Nets in Base 2

(66−33, 66, 24)-Net over F₂ — Constructive and digital

Digital (33, 66, 24)-net over F₂, using

net from sequence [i] based on digital (33, 23)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
  - a function field by SÃ©mirat [i]

(66−33, 66, 28)-Net over F₂ — Digital

Digital (33, 66, 28)-net over F₂, using

net from sequence [i] based on digital (33, 27)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 28, using
  - Drinfeld modules of rank 1 [i]

(66−33, 66, 75)-Net over F₂ — Upper bound on s (digital)

There is no digital (33, 66, 76)-net over F₂, because

1 times m-reduction [i] would yield digital (33, 65, 76)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁶⁵, 76, F₂, 32) (dual of [76, 11, 33]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(2⁶⁴, 72, F₂, 32) (dual of [72, 8, 33]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁶⁵, 73, F₂, 33) (dual of [73, 8, 34]-code), but
        â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(2¹¹, 76, S₂, 4), but
      - discarding factors would yield OA(2¹¹, 64, S₂, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 2081 > 2¹¹ [i]

(66−33, 66, 80)-Net in Base 2 — Upper bound on s

There is no (33, 66, 81)-net in base 2, because

1 times m-reduction [i] would yield (33, 65, 81)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁵, 81, S₂, 32), but
  - the linear programming bound shows that MÂ â‰¥ 648144 799773 858805 579776 / 14875 > 2⁶⁵ [i]