MinT - Best Known (235−107, 235, s)-Nets in Base 2

Best Known (235−107, 235, s)-Nets in Base 2

(235−107, 235, 63)-Net over F₂ — Constructive and digital

Digital (128, 235, 63)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 74, 21)-net over F₂, using
  - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 161, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(235−107, 235, 81)-Net over F₂ — Digital

Digital (128, 235, 81)-net over F₂, using

t-expansion [i] based on digital (126, 235, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(235−107, 235, 285)-Net in Base 2 — Upper bound on s

There is no (128, 235, 286)-net in base 2, because

1 times m-reduction [i] would yield (128, 234, 286)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁴, 286, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 107485 368431 143344 520330 133388 459799 802744 525415 015694 365229 341054 824013 444086 208158 236672 / 3 580754 389815 658575 > 2²³⁴ [i]