MinT - Best Known (240−111, 240, s)-Nets in Base 2

Best Known (240−111, 240, s)-Nets in Base 2

(240−111, 240, 62)-Net over F₂ — Constructive and digital

Digital (129, 240, 62)-net over F₂, using

1 times m-reduction [i] based on digital (129, 241, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 75, 20)-net over F₂, using
    - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 166, 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]

(240−111, 240, 81)-Net over F₂ — Digital

Digital (129, 240, 81)-net over F₂, using

t-expansion [i] based on digital (126, 240, 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]

(240−111, 240, 284)-Net in Base 2 — Upper bound on s

There is no (129, 240, 285)-net in base 2, because

3 times m-reduction [i] would yield (129, 237, 285)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁷, 285, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 56343 399996 195265 169227 114420 218575 369487 305782 122681 969429 903696 483971 902670 896111 812608 / 204485 216198 046875 > 2²³⁷ [i]