MinT - Best Known (170−117, 170, s)-Nets in Base 8

Best Known (170−117, 170, s)-Nets in Base 8

(170−117, 170, 98)-Net over F₈ — Constructive and digital

Digital (53, 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]

(170−117, 170, 144)-Net over F₈ — Digital

Digital (53, 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]

(170−117, 170, 1336)-Net in Base 8 — Upper bound on s

There is no (53, 170, 1337)-net in base 8, because

1 times m-reduction [i] would yield (53, 169, 1337)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 434 914640 450216 849974 581917 830832 982867 379628 204955 341815 906804 365132 669179 344349 106546 939292 063917 686465 727343 293381 060694 401707 680418 807515 458537 011904 > 8¹⁶⁹ [i]