MinT - Best Known (251−142, 251, s)-Nets in Base 2

Best Known (251−142, 251, s)-Nets in Base 2

(251−142, 251, 56)-Net over F₂ — Constructive and digital

Digital (109, 251, 56)-net over F₂, using

t-expansion [i] based on digital (105, 251, 56)-net over F₂, using
- net from sequence [i] based on digital (105, 55)-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 7 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]

(251−142, 251, 65)-Net over F₂ — Digital

Digital (109, 251, 65)-net over F₂, using

t-expansion [i] based on digital (95, 251, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(251−142, 251, 223)-Net in Base 2 — Upper bound on s

There is no (109, 251, 224)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3692 199356 891555 853069 471176 457465 430220 710535 084256 735204 236934 440905 412995 > 2²⁵¹ [i]