MinT - Best Known (138−85, 138, s)-Nets in Base 2

Best Known (138−85, 138, s)-Nets in Base 2

(138−85, 138, 36)-Net over F₂ — Constructive and digital

Digital (53, 138, 36)-net over F₂, using

t-expansion [i] based on digital (51, 138, 36)-net over F₂, using
- net from sequence [i] based on digital (51, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(138−85, 138, 40)-Net over F₂ — Digital

Digital (53, 138, 40)-net over F₂, using

t-expansion [i] based on digital (50, 138, 40)-net over F₂, using
- net from sequence [i] based on digital (50, 39)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 50 and N(F) ≥ 40, using
    - Drinfeld modules of rank 1 [i]

(138−85, 138, 105)-Net in Base 2 — Upper bound on s

There is no (53, 138, 106)-net in base 2, because

1 times m-reduction [i] would yield (53, 137, 106)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 215511 310209 619691 349360 192869 992416 246200 > 2¹³⁷ [i]