MinT - Best Known (56, 56+103, s)-Nets in Base 8

Best Known (56, 56+103, s)-Nets in Base 8

(56, 56+103, 98)-Net over F₈ — Constructive and digital

Digital (56, 159, 98)-net over F₈, using

t-expansion [i] based on digital (37, 159, 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+103, 144)-Net over F₈ — Digital

Digital (56, 159, 144)-net over F₈, using

t-expansion [i] based on digital (45, 159, 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+103, 1748)-Net in Base 8 — Upper bound on s

There is no (56, 159, 1749)-net in base 8, because

1 times m-reduction [i] would yield (56, 158, 1749)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 49496 563544 338605 170846 667451 282691 939762 761639 728718 190559 703138 937415 487704 808576 090238 338196 725676 204459 715737 584647 769353 902502 766286 912424 > 8¹⁵⁸ [i]