MinT - Best Known (68, 68+97, s)-Nets in Base 8

Best Known (68, 68+97, s)-Nets in Base 8

(68, 68+97, 98)-Net over F₈ — Constructive and digital

Digital (68, 165, 98)-net over F₈, using

t-expansion [i] based on digital (37, 165, 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]

(68, 68+97, 144)-Net over F₈ — Digital

Digital (68, 165, 144)-net over F₈, using

t-expansion [i] based on digital (45, 165, 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]

(68, 68+97, 150)-Net in Base 8

(68, 165, 150)-net in base 8, using

3 times m-reduction [i] based on (68, 168, 150)-net in base 8, using
- base change [i] based on digital (26, 126, 150)-net over F₁₆, using
  - net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
      - a curve from the manYPoints database [i]

(68, 68+97, 3229)-Net in Base 8 — Upper bound on s

There is no (68, 165, 3230)-net in base 8, because

1 times m-reduction [i] would yield (68, 164, 3230)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12793 175904 548377 618948 225252 036862 113539 940828 499920 704682 497389 491185 998940 158613 652164 631298 819945 244763 168110 034137 383493 298429 397548 897573 647356 > 8¹⁶⁴ [i]