MinT - Best Known (147, 147+91, s)-Nets in Base 2

Best Known (147, 147+91, s)-Nets in Base 2

(147, 147+91, 77)-Net over F₂ — Constructive and digital

Digital (147, 238, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (48, 93, 35)-net over F₂, using
  - net from sequence [i] based on digital (48, 34)-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 2 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, 145, 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]

(147, 147+91, 84)-Net in Base 2 — Constructive

(147, 238, 84)-net in base 2, using

2 times m-reduction [i] based on (147, 240, 84)-net in base 2, using
- trace code for nets [i] based on (27, 120, 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]

(147, 147+91, 119)-Net over F₂ — Digital

Digital (147, 238, 119)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(147, 147+91, 614)-Net in Base 2 — Upper bound on s

There is no (147, 238, 615)-net in base 2, because

1 times m-reduction [i] would yield (147, 237, 615)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 229095 435135 988798 165815 913780 481082 815433 376589 654765 263039 155284 283744 > 2²³⁷ [i]