MinT - Best Known (42, 135, s)-Nets in Base 8

Best Known (42, 135, s)-Nets in Base 8

(42, 135, 98)-Net over F₈ — Constructive and digital

Digital (42, 135, 98)-net over F₈, using

t-expansion [i] based on digital (37, 135, 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]

(42, 135, 129)-Net over F₈ — Digital

Digital (42, 135, 129)-net over F₈, using

t-expansion [i] based on digital (38, 135, 129)-net over F₈, using
- net from sequence [i] based on digital (38, 128)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(42, 135, 1069)-Net in Base 8 — Upper bound on s

There is no (42, 135, 1070)-net in base 8, because

1 times m-reduction [i] would yield (42, 134, 1070)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 397425 958618 118499 230405 417415 165560 390195 631184 984873 492082 924573 500794 868556 135049 818907 205537 414125 978277 514478 347216 > 8¹³⁴ [i]