MinT - Best Known (53, 53+85, s)-Nets in Base 8

Best Known (53, 53+85, s)-Nets in Base 8

(53, 53+85, 98)-Net over F₈ — Constructive and digital

Digital (53, 138, 98)-net over F₈, using

t-expansion [i] based on digital (37, 138, 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]

(53, 53+85, 144)-Net over F₈ — Digital

Digital (53, 138, 144)-net over F₈, using

t-expansion [i] based on digital (45, 138, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(53, 53+85, 2055)-Net in Base 8 — Upper bound on s

There is no (53, 138, 2056)-net in base 8, because

1 times m-reduction [i] would yield (53, 137, 2056)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5322 484898 752046 183402 633934 917193 210975 425671 783472 851268 005945 617618 882127 839201 660894 263072 460555 814207 853193 889669 732608 > 8¹³⁷ [i]