MinT - Best Known (85−47, 85, s)-Nets in Base 2

Best Known (85−47, 85, s)-Nets in Base 2

(85−47, 85, 24)-Net over F₂ — Constructive and digital

Digital (38, 85, 24)-net over F₂, using

t-expansion [i] based on digital (33, 85, 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]

(85−47, 85, 30)-Net over F₂ — Digital

Digital (38, 85, 30)-net over F₂, using

t-expansion [i] based on digital (36, 85, 30)-net over F₂, using
- net from sequence [i] based on digital (36, 29)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
    - Drinfeld modules of rank 1 [i]

(85−47, 85, 84)-Net over F₂ — Upper bound on s (digital)

There is no digital (38, 85, 85)-net over F₂, because

7 times m-reduction [i] would yield digital (38, 78, 85)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷⁸, 85, F₂, 40) (dual of [85, 7, 41]-code), but
  - â€œHelâ€ bound on codes from Brouwerâ€™s database [i]

(85−47, 85, 86)-Net in Base 2 — Upper bound on s

There is no (38, 85, 87)-net in base 2, because

3 times m-reduction [i] would yield (38, 82, 87)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁸², 87, S₂, 44), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 77 371252 455336 267181 195264 / 15 > 2⁸² [i]