MinT - Best Known (237−107, 237, s)-Nets in Base 2

Best Known (237−107, 237, s)-Nets in Base 2

(237−107, 237, 63)-Net over F₂ — Constructive and digital

Digital (130, 237, 63)-net over F₂, using

3 times m-reduction [i] based on digital (130, 240, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 76, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 164, 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]

(237−107, 237, 81)-Net over F₂ — Digital

Digital (130, 237, 81)-net over F₂, using

t-expansion [i] based on 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]

(237−107, 237, 293)-Net in Base 2 — Upper bound on s

There is no (130, 237, 294)-net in base 2, because

1 times m-reduction [i] would yield (130, 236, 294)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁶, 294, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 355 957995 952588 693183 395284 979830 322852 701354 338332 589389 465361 179247 273153 986172 267263 950848 / 2810 579332 872546 572337 > 2²³⁶ [i]