MinT - Best Known (240−135, 240, s)-Nets in Base 2

Best Known (240−135, 240, s)-Nets in Base 2

(240−135, 240, 56)-Net over F₂ — Constructive and digital

Digital (105, 240, 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]

(240−135, 240, 65)-Net over F₂ — Digital

Digital (105, 240, 65)-net over F₂, using

t-expansion [i] based on digital (95, 240, 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]

(240−135, 240, 218)-Net in Base 2 — Upper bound on s

There is no (105, 240, 219)-net in base 2, because

1 times m-reduction [i] would yield (105, 239, 219)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 016362 939944 127166 336947 963786 899550 549880 186486 110073 349332 538836 233378 > 2²³⁹ [i]