MinT - Best Known (126, 126+67, s)-Nets in Base 2

Best Known (126, 126+67, s)-Nets in Base 2

(126, 126+67, 75)-Net over F₂ — Constructive and digital

Digital (126, 193, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 72, 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, 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]

(126, 126+67, 84)-Net in Base 2 — Constructive

(126, 193, 84)-net in base 2, using

5 times m-reduction [i] based on (126, 198, 84)-net in base 2, using
- trace code for nets [i] based on (27, 99, 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]

(126, 126+67, 124)-Net over F₂ — Digital

Digital (126, 193, 124)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+67, 695)-Net in Base 2 — Upper bound on s

There is no (126, 193, 696)-net in base 2, because

1 times m-reduction [i] would yield (126, 192, 696)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6531 615170 648894 916554 041363 855014 370308 052368 132744 443427 > 2¹⁹² [i]