MinT - Best Known (247−94, 247, s)-Nets in Base 2

Best Known (247−94, 247, s)-Nets in Base 2

(247−94, 247, 78)-Net over F₂ — Constructive and digital

Digital (153, 247, 78)-net over F₂, using

2 times m-reduction [i] based on digital (153, 249, 78)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (51, 99, 36)-net over F₂, using
    - net from sequence [i] based on digital (51, 35)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
  2. digital (54, 150, 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]

(247−94, 247, 84)-Net in Base 2 — Constructive

(153, 247, 84)-net in base 2, using

5 times m-reduction [i] based on (153, 252, 84)-net in base 2, using
- trace code for nets [i] based on (27, 126, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [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]

(247−94, 247, 124)-Net over F₂ — Digital

Digital (153, 247, 124)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−94, 247, 634)-Net in Base 2 — Upper bound on s

There is no (153, 247, 635)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 231 342927 822378 102297 109629 154253 006130 527861 528008 362737 077219 494719 045896 > 2²⁴⁷ [i]