MinT - Best Known (135, 135+107, s)-Nets in Base 2

Best Known (135, 135+107, s)-Nets in Base 2

(135, 135+107, 66)-Net over F₂ — Constructive and digital

Digital (135, 242, 66)-net over F₂, using

7 times m-reduction [i] based on digital (135, 249, 66)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 96, 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 (39, 153, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(135, 135+107, 86)-Net over F₂ — Digital

Digital (135, 242, 86)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+107, 407)-Net in Base 2 — Upper bound on s

There is no (135, 242, 408)-net in base 2, because

1 times m-reduction [i] would yield (135, 241, 408)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 542181 220824 562260 871678 626741 237576 565568 506564 064855 080331 990622 553604 > 2²⁴¹ [i]