MinT - Best Known (53, 53+89, s)-Nets in Base 8

Best Known (53, 53+89, s)-Nets in Base 8

(53, 53+89, 98)-Net over F₈ — Constructive and digital

Digital (53, 142, 98)-net over F₈, using

t-expansion [i] based on digital (37, 142, 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+89, 144)-Net over F₈ — Digital

Digital (53, 142, 144)-net over F₈, using

t-expansion [i] based on digital (45, 142, 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+89, 1903)-Net in Base 8 — Upper bound on s

There is no (53, 142, 1904)-net in base 8, because

1 times m-reduction [i] would yield (53, 141, 1904)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 913495 882441 006784 205825 885274 423001 467167 745610 061313 358272 909077 165401 145576 495854 693224 540692 021932 011171 294516 559309 333625 > 8¹⁴¹ [i]