MinT - Best Known (149−45, 149, s)-Nets in Base 2

Best Known (149−45, 149, s)-Nets in Base 2

(149−45, 149, 76)-Net over F₂ — Constructive and digital

Digital (104, 149, 76)-net over F₂, using

2¹ times duplication [i] based on digital (103, 148, 76)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 28, 10)-net over F₂, using
    - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
        an explicitly constructive algebraic function field [i]
    - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
  2. digital (75, 120, 66)-net over F₂, using
    - trace code for nets [i] based on digital (15, 60, 33)-net over F₄, using
      - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]

(149−45, 149, 84)-Net in Base 2 — Constructive

(104, 149, 84)-net in base 2, using

5 times m-reduction [i] based on (104, 154, 84)-net in base 2, using
- trace code for nets [i] based on (27, 77, 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]

(149−45, 149, 138)-Net over F₂ — Digital

Digital (104, 149, 138)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(149−45, 149, 927)-Net in Base 2 — Upper bound on s

There is no (104, 149, 928)-net in base 2, because

1 times m-reduction [i] would yield (104, 148, 928)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 362 472559 455426 934664 599588 563418 935262 638083 > 2¹⁴⁸ [i]