MinT - Best Known (237−111, 237, s)-Nets in Base 2

Best Known (237−111, 237, s)-Nets in Base 2

(237−111, 237, 59)-Net over F₂ — Constructive and digital

Digital (126, 237, 59)-net over F₂, using

3 times m-reduction [i] based on digital (126, 240, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 72, 17)-net over F₂, using
    - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 168, 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]

(237−111, 237, 81)-Net over F₂ — Digital

Digital (126, 237, 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]

(237−111, 237, 274)-Net in Base 2 — Upper bound on s

There is no (126, 237, 275)-net in base 2, because

3 times m-reduction [i] would yield (126, 234, 275)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁴, 275, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 275 709011 416343 386283 194427 515396 341112 620119 523157 025502 536288 617777 395593 653660 942336 / 9364 570416 997125 > 2²³⁴ [i]