MinT - Best Known (81, 81+109, s)-Nets in Base 2

Best Known (81, 81+109, s)-Nets in Base 2

(81, 81+109, 51)-Net over F₂ — Constructive and digital

Digital (81, 190, 51)-net over F₂, using

t-expansion [i] based on digital (80, 190, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-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, 1 place with degree 2, and 2 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(81, 81+109, 56)-Net over F₂ — Digital

Digital (81, 190, 56)-net over F₂, using

t-expansion [i] based on digital (80, 190, 56)-net over F₂, using
- net from sequence [i] based on digital (80, 55)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 80 and N(F) ≥ 56, using
    - Drinfeld modules of rank 1 [i]

(81, 81+109, 167)-Net in Base 2 — Upper bound on s

There is no (81, 190, 168)-net in base 2, because

1 times m-reduction [i] would yield (81, 189, 168)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 923 940887 800161 557236 416440 849976 056892 856965 755031 924472 > 2¹⁸⁹ [i]