MinT - Best Known (188−121, 188, s)-Nets in Base 2

Best Known (188−121, 188, s)-Nets in Base 2

(188−121, 188, 43)-Net over F₂ — Constructive and digital

Digital (67, 188, 43)-net over F₂, using

t-expansion [i] based on digital (59, 188, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(188−121, 188, 48)-Net over F₂ — Digital

Digital (67, 188, 48)-net over F₂, using

t-expansion [i] based on digital (65, 188, 48)-net over F₂, using
- net from sequence [i] based on digital (65, 47)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 65 and N(F) ≥ 48, using
    - Drinfeld modules of rank 1 [i]

(188−121, 188, 126)-Net in Base 2 — Upper bound on s

There is no (67, 188, 127)-net in base 2, because

1 times m-reduction [i] would yield (67, 187, 127)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 233 124449 784377 252402 689572 825342 544143 641675 444225 986001 > 2¹⁸⁷ [i]