MinT - Best Known (56, 56+89, s)-Nets in Base 8

Best Known (56, 56+89, s)-Nets in Base 8

(56, 56+89, 98)-Net over F₈ — Constructive and digital

Digital (56, 145, 98)-net over F₈, using

t-expansion [i] based on digital (37, 145, 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]

(56, 56+89, 144)-Net over F₈ — Digital

Digital (56, 145, 144)-net over F₈, using

t-expansion [i] based on digital (45, 145, 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]

(56, 56+89, 2197)-Net in Base 8 — Upper bound on s

There is no (56, 145, 2198)-net in base 8, because

1 times m-reduction [i] would yield (56, 144, 2198)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11170 685374 810643 592578 629702 366282 087579 652494 409577 740790 887291 430616 408315 179461 220568 339273 832985 765478 095150 501354 881091 199128 > 8¹⁴⁴ [i]