MinT - Best Known (215−83, 215, s)-Nets in Base 2

Best Known (215−83, 215, s)-Nets in Base 2

(215−83, 215, 70)-Net over F₂ — Constructive and digital

Digital (132, 215, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 62, 21)-net over F₂, using
  - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 153, 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]

(215−83, 215, 106)-Net over F₂ — Digital

Digital (132, 215, 106)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−83, 215, 543)-Net in Base 2 — Upper bound on s

There is no (132, 215, 544)-net in base 2, because

1 times m-reduction [i] would yield (132, 214, 544)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 28087 839021 478334 517489 261352 951802 364590 092725 266592 304098 665040 > 2²¹⁴ [i]