MinT - Best Known (125, 125+67, s)-Nets in Base 2

Best Known (125, 125+67, s)-Nets in Base 2

(125, 125+67, 70)-Net over F₂ — Constructive and digital

Digital (125, 192, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (38, 71, 28)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (11, 27, 14)-net over F₂, using
      - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
        an explicitly constructive algebraic function field [i]
      - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
    2. digital (11, 44, 14)-net over F₂, using
      - net from sequence [i] based on digital (11, 13)-sequence over F₂ (see above)
2. digital (54, 121, 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]

(125, 125+67, 84)-Net in Base 2 — Constructive

(125, 192, 84)-net in base 2, using

4 times m-reduction [i] based on (125, 196, 84)-net in base 2, using
- trace code for nets [i] based on (27, 98, 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]

(125, 125+67, 122)-Net over F₂ — Digital

Digital (125, 192, 122)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+67, 679)-Net in Base 2 — Upper bound on s

There is no (125, 192, 680)-net in base 2, because

1 times m-reduction [i] would yield (125, 191, 680)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3181 803180 983577 838837 398811 532425 854115 593653 812636 043853 > 2¹⁹¹ [i]