MinT - Best Known (148−45, 148, s)-Nets in Base 2

Best Known (148−45, 148, s)-Nets in Base 2

(148−45, 148, 76)-Net over F₂ — Constructive and digital

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]

(148−45, 148, 84)-Net in Base 2 — Constructive

(103, 148, 84)-net in base 2, using

4 times m-reduction [i] based on (103, 152, 84)-net in base 2, using
- trace code for nets [i] based on (27, 76, 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]

(148−45, 148, 135)-Net over F₂ — Digital

Digital (103, 148, 135)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(148−45, 148, 897)-Net in Base 2 — Upper bound on s

There is no (103, 148, 898)-net in base 2, because

1 times m-reduction [i] would yield (103, 147, 898)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 180 208465 838330 127260 252267 415845 058523 043648 > 2¹⁴⁷ [i]