MinT - Best Known (135, 135+79, s)-Nets in Base 2

Best Known (135, 135+79, s)-Nets in Base 2

(135, 135+79, 75)-Net over F₂ — Constructive and digital

Digital (135, 214, 75)-net over F₂, using

5 times m-reduction [i] based on digital (135, 219, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 81, 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 (54, 138, 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]

(135, 135+79, 84)-Net in Base 2 — Constructive

(135, 214, 84)-net in base 2, using

2 times m-reduction [i] based on (135, 216, 84)-net in base 2, using
- trace code for nets [i] based on (27, 108, 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]

(135, 135+79, 117)-Net over F₂ — Digital

Digital (135, 214, 117)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+79, 622)-Net in Base 2 — Upper bound on s

There is no (135, 214, 623)-net in base 2, because

1 times m-reduction [i] would yield (135, 213, 623)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13463 948891 170796 870217 649243 324474 676966 513413 814096 639934 863014 > 2²¹³ [i]