MinT - Best Known (194−70, 194, s)-Nets in Base 2

Best Known (194−70, 194, s)-Nets in Base 2

(194−70, 194, 69)-Net over F₂ — Constructive and digital

Digital (124, 194, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 54, 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 (70, 140, 49)-net over F₂, using
  - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(194−70, 194, 84)-Net in Base 2 — Constructive

(124, 194, 84)-net in base 2, using

trace code for nets [i] based on (27, 97, 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]

(194−70, 194, 114)-Net over F₂ — Digital

Digital (124, 194, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(194−70, 194, 598)-Net in Base 2 — Upper bound on s

There is no (124, 194, 599)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 26207 188967 554199 990825 963488 003018 021589 534073 774669 292380 > 2¹⁹⁴ [i]