MinT - Best Known (87−45, 87, s)-Nets in Base 8

Best Known (87−45, 87, s)-Nets in Base 8

(87−45, 87, 98)-Net over F₈ — Constructive and digital

Digital (42, 87, 98)-net over F₈, using

t-expansion [i] based on digital (37, 87, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(87−45, 87, 136)-Net over F₈ — Digital

Digital (42, 87, 136)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−45, 87, 4371)-Net in Base 8 — Upper bound on s

There is no (42, 87, 4372)-net in base 8, because

1 times m-reduction [i] would yield (42, 86, 4372)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 463546 558337 672890 007754 722947 863472 882451 610820 041269 367554 764942 824134 019776 > 8⁸⁶ [i]