MinT - Best Known (247−117, 247, s)-Nets in Base 2

Best Known (247−117, 247, s)-Nets in Base 2

(247−117, 247, 59)-Net over F₂ — Constructive and digital

Digital (130, 247, 59)-net over F₂, using

5 times m-reduction [i] based on digital (130, 252, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 76, 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, 176, 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]

(247−117, 247, 81)-Net over F₂ — Digital

Digital (130, 247, 81)-net over F₂, using

t-expansion [i] based on digital (126, 247, 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]

(247−117, 247, 286)-Net in Base 2 — Upper bound on s

There is no (130, 247, 287)-net in base 2, because

9 times m-reduction [i] would yield (130, 238, 287)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁸, 287, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 2 133520 632200 595168 112459 587583 515002 010526 895398 363193 388655 973626 286863 730797 742512 930816 / 4 628313 007677 734375 > 2²³⁸ [i]