MinT - Best Known (220−79, 220, s)-Nets in Base 2

Best Known (220−79, 220, s)-Nets in Base 2

(220−79, 220, 77)-Net over F₂ — Constructive and digital

Digital (141, 220, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (48, 87, 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, 133, 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]

(220−79, 220, 86)-Net in Base 2 — Constructive

(141, 220, 86)-net in base 2, using

2 times m-reduction [i] based on (141, 222, 86)-net in base 2, using
- trace code for nets [i] based on (30, 111, 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]

(220−79, 220, 128)-Net over F₂ — Digital

Digital (141, 220, 128)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−79, 220, 698)-Net in Base 2 — Upper bound on s

There is no (141, 220, 699)-net in base 2, because

1 times m-reduction [i] would yield (141, 219, 699)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 853691 990565 339081 986533 903719 096632 508382 343881 394382 642954 100068 > 2²¹⁹ [i]