MinT - Best Known (126, 126+69, s)-Nets in Base 2

Best Known (126, 126+69, s)-Nets in Base 2

(126, 126+69, 70)-Net over F₂ — Constructive and digital

Digital (126, 195, 70)-net over F₂, using

1 times m-reduction [i] based on digital (126, 196, 70)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 56, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (70, 140, 49)-net over F₂, using
    - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(126, 126+69, 84)-Net in Base 2 — Constructive

(126, 195, 84)-net in base 2, using

3 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+69, 119)-Net over F₂ — Digital

Digital (126, 195, 119)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+69, 657)-Net in Base 2 — Upper bound on s

There is no (126, 195, 658)-net in base 2, because

1 times m-reduction [i] would yield (126, 194, 658)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 25600 760305 874783 599475 154543 720125 184708 699457 874562 151936 > 2¹⁹⁴ [i]