MinT - Best Known (164−103, 164, s)-Nets in Base 8

Best Known (164−103, 164, s)-Nets in Base 8

(164−103, 164, 98)-Net over F₈ — Constructive and digital

Digital (61, 164, 98)-net over F₈, using

t-expansion [i] based on digital (37, 164, 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]

(164−103, 164, 144)-Net over F₈ — Digital

Digital (61, 164, 144)-net over F₈, using

t-expansion [i] based on digital (45, 164, 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]

(164−103, 164, 2151)-Net in Base 8 — Upper bound on s

There is no (61, 164, 2152)-net in base 8, because

1 times m-reduction [i] would yield (61, 163, 2152)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1631 835784 709828 823275 929243 314181 898239 060101 032201 748890 280406 186669 040659 832235 489093 807859 303660 534530 692520 871765 091317 900102 891358 358830 647740 > 8¹⁶³ [i]