MinT - Best Known (171−103, 171, s)-Nets in Base 8

Best Known (171−103, 171, s)-Nets in Base 8

(171−103, 171, 98)-Net over F₈ — Constructive and digital

Digital (68, 171, 98)-net over F₈, using

t-expansion [i] based on digital (37, 171, 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]

(171−103, 171, 144)-Net over F₈ — Digital

Digital (68, 171, 144)-net over F₈, using

t-expansion [i] based on digital (45, 171, 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]

(171−103, 171, 2872)-Net in Base 8 — Upper bound on s

There is no (68, 171, 2873)-net in base 8, because

1 times m-reduction [i] would yield (68, 170, 2873)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3397 501231 835768 756349 644004 724929 380834 135903 269855 697332 868823 663513 478880 159367 996508 146565 775912 127249 106118 941279 404856 092699 923138 091381 569778 084792 > 8¹⁷⁰ [i]