MinT - Best Known (126, 126+104, s)-Nets in Base 2

Best Known (126, 126+104, s)-Nets in Base 2

(126, 126+104, 62)-Net over F₂ — Constructive and digital

Digital (126, 230, 62)-net over F₂, using

2 times m-reduction [i] based on digital (126, 232, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 72, 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, 160, 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]

(126, 126+104, 81)-Net over F₂ — Digital

Digital (126, 230, 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]

(126, 126+104, 282)-Net in Base 2 — Upper bound on s

There is no (126, 230, 283)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²³⁰, 283, S₂, 104), but
- adding a parity check bit [i] would yield OA(2²³¹, 284, S₂, 105), but
  - the linear programming bound shows that MÂ â‰¥ 208557 008507 723789 031557 389329 381561 817137 556188 539558 960408 788198 339282 963223 421267 214336 / 47 001142 383583 644621 > 2²³¹ [i]