MinT - Best Known (133, 133+71, s)-Nets in Base 2

Best Known (133, 133+71, s)-Nets in Base 2

(133, 133+71, 76)-Net over F₂ — Constructive and digital

Digital (133, 204, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 74, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (59, 130, 43)-net over F₂, using
  - net from sequence [i] based on digital (59, 42)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(133, 133+71, 86)-Net in Base 2 — Constructive

(133, 204, 86)-net in base 2, using

2 times m-reduction [i] based on (133, 206, 86)-net in base 2, using
- trace code for nets [i] based on (30, 103, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(133, 133+71, 129)-Net over F₂ — Digital

Digital (133, 204, 129)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(133, 133+71, 724)-Net in Base 2 — Upper bound on s

There is no (133, 204, 725)-net in base 2, because

1 times m-reduction [i] would yield (133, 203, 725)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13 193585 873549 752775 271760 359084 681578 241624 921893 705621 143360 > 2²⁰³ [i]