MinT - Best Known (78−41, 78, s)-Nets in Base 2

Best Known (78−41, 78, s)-Nets in Base 2

(78−41, 78, 24)-Net over F₂ — Constructive and digital

Digital (37, 78, 24)-net over F₂, using

t-expansion [i] based on digital (33, 78, 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]

(78−41, 78, 30)-Net over F₂ — Digital

Digital (37, 78, 30)-net over F₂, using

t-expansion [i] based on digital (36, 78, 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]

(78−41, 78, 82)-Net over F₂ — Upper bound on s (digital)

There is no digital (37, 78, 83)-net over F₂, because

1 times m-reduction [i] would yield digital (37, 77, 83)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷⁷, 83, F₂, 40) (dual of [83, 6, 41]-code), but
  - Griesmer bound [i]

(78−41, 78, 85)-Net in Base 2 — Upper bound on s

There is no (37, 78, 86)-net in base 2, because

3 times m-reduction [i] would yield (37, 75, 86)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷⁵, 86, S₂, 38), but
  - the linear programming bound shows that MÂ â‰¥ 6 120186 961799 060196 950016 / 125 > 2⁷⁵ [i]