MinT - Best Known (210−131, 210, s)-Nets in Base 2

Best Known (210−131, 210, s)-Nets in Base 2

(210−131, 210, 50)-Net over F₂ — Constructive and digital

Digital (79, 210, 50)-net over F₂, using

t-expansion [i] based on digital (75, 210, 50)-net over F₂, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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]

(210−131, 210, 52)-Net over F₂ — Digital

Digital (79, 210, 52)-net over F₂, using

t-expansion [i] based on digital (77, 210, 52)-net over F₂, using
- net from sequence [i] based on digital (77, 51)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 77 and N(F) ≥ 52, using
    - Drinfeld modules of rank 1 [i]

(210−131, 210, 150)-Net in Base 2 — Upper bound on s

There is no (79, 210, 151)-net in base 2, because

1 times m-reduction [i] would yield (79, 209, 151)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 870 602377 415959 636044 967284 385145 922195 096317 616991 507246 017744 > 2²⁰⁹ [i]