MinT - Best Known (231−67, 231, s)-Nets in Base 2

Best Known (231−67, 231, s)-Nets in Base 2

(231−67, 231, 138)-Net over F₂ — Constructive and digital

Digital (164, 231, 138)-net over F₂, using

trace code for nets [i] based on digital (10, 77, 46)-net over F₈, using
- net from sequence [i] based on digital (10, 45)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]

(231−67, 231, 219)-Net over F₂ — Digital

Digital (164, 231, 219)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−67, 231, 1601)-Net in Base 2 — Upper bound on s

There is no (164, 231, 1602)-net in base 2, because

1 times m-reduction [i] would yield (164, 230, 1602)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1742 910171 972451 076656 163334 894479 008412 949720 476238 704423 928676 723330 > 2²³⁰ [i]