MinT - Best Known (56, 56+83, s)-Nets in Base 8

Best Known (56, 56+83, s)-Nets in Base 8

(56, 56+83, 98)-Net over F₈ — Constructive and digital

Digital (56, 139, 98)-net over F₈, using

t-expansion [i] based on digital (37, 139, 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+83, 144)-Net over F₈ — Digital

Digital (56, 139, 144)-net over F₈, using

t-expansion [i] based on digital (45, 139, 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+83, 2500)-Net in Base 8 — Upper bound on s

There is no (56, 139, 2501)-net in base 8, because

1 times m-reduction [i] would yield (56, 138, 2501)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42641 536773 235393 170585 556367 909387 664826 212581 194657 968359 309457 006621 948580 342037 883017 856559 043186 001032 022918 264354 899808 > 8¹³⁸ [i]