MinT - Best Known (170−105, 170, s)-Nets in Base 8

Best Known (170−105, 170, s)-Nets in Base 8

(170−105, 170, 98)-Net over F₈ — Constructive and digital

Digital (65, 170, 98)-net over F₈, using

t-expansion [i] based on digital (37, 170, 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]

(170−105, 170, 144)-Net over F₈ — Digital

Digital (65, 170, 144)-net over F₈, using

t-expansion [i] based on digital (45, 170, 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]

(170−105, 170, 2455)-Net in Base 8 — Upper bound on s

There is no (65, 170, 2456)-net in base 8, because

1 times m-reduction [i] would yield (65, 169, 2456)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 426 484283 049812 999417 457293 681455 825831 630012 077952 758070 954184 409838 429604 297261 862391 435248 843995 114613 186665 152482 497667 544649 550174 465311 038205 897508 > 8¹⁶⁹ [i]