MinT - Best Known (230−107, 230, s)-Nets in Base 2

Best Known (230−107, 230, s)-Nets in Base 2

(230−107, 230, 59)-Net over F₂ — Constructive and digital

Digital (123, 230, 59)-net over F₂, using

1 times m-reduction [i] based on digital (123, 231, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 69, 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, 162, 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]

(230−107, 230, 80)-Net over F₂ — Digital

Digital (123, 230, 80)-net over F₂, using

t-expansion [i] based on digital (121, 230, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(230−107, 230, 266)-Net in Base 2 — Upper bound on s

There is no (123, 230, 267)-net in base 2, because

1 times m-reduction [i] would yield (123, 229, 267)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁹, 267, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 5463 389198 461980 719621 493892 918669 384263 874368 982752 322932 717321 849363 217518 690304 / 5 086453 978125 > 2²²⁹ [i]