MinT - Best Known (238−109, 238, s)-Nets in Base 2

Best Known (238−109, 238, s)-Nets in Base 2

(238−109, 238, 63)-Net over F₂ — Constructive and digital

Digital (129, 238, 63)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 75, 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, 163, 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]

(238−109, 238, 81)-Net over F₂ — Digital

Digital (129, 238, 81)-net over F₂, using

t-expansion [i] based on digital (126, 238, 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]

(238−109, 238, 284)-Net in Base 2 — Upper bound on s

There is no (129, 238, 285)-net in base 2, because

1 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]