MinT - Best Known (234−90, 234, s)-Nets in Base 2

Best Known (234−90, 234, s)-Nets in Base 2

(234−90, 234, 76)-Net over F₂ — Constructive and digital

Digital (144, 234, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (45, 90, 34)-net over F₂, using
  - net from sequence [i] based on digital (45, 33)-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 1 place 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, 144, 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]

(234−90, 234, 84)-Net in Base 2 — Constructive

(144, 234, 84)-net in base 2, using

trace code for nets [i] based on (27, 117, 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]

(234−90, 234, 115)-Net over F₂ — Digital

Digital (144, 234, 115)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−90, 234, 583)-Net in Base 2 — Upper bound on s

There is no (144, 234, 584)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27733 136429 497560 878811 955898 222020 256422 568044 869352 441701 058001 146120 > 2²³⁴ [i]