MinT - Best Known (81, 81+107, s)-Nets in Base 2

Best Known (81, 81+107, s)-Nets in Base 2

(81, 81+107, 51)-Net over F₂ — Constructive and digital

Digital (81, 188, 51)-net over F₂, using

t-expansion [i] based on digital (80, 188, 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+107, 56)-Net over F₂ — Digital

Digital (81, 188, 56)-net over F₂, using

t-expansion [i] based on digital (80, 188, 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+107, 168)-Net in Base 2 — Upper bound on s

There is no (81, 188, 169)-net in base 2, because

1 times m-reduction [i] would yield (81, 187, 169)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 203 146822 495693 423946 499114 389269 630399 116231 397922 431904 > 2¹⁸⁷ [i]