MinT - Best Known (52, 52+97, s)-Nets in Base 8

Best Known (52, 52+97, s)-Nets in Base 8

(52, 52+97, 98)-Net over F₈ — Constructive and digital

Digital (52, 149, 98)-net over F₈, using

t-expansion [i] based on digital (37, 149, 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]

(52, 52+97, 144)-Net over F₈ — Digital

Digital (52, 149, 144)-net over F₈, using

t-expansion [i] based on digital (45, 149, 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]

(52, 52+97, 1599)-Net in Base 8 — Upper bound on s

There is no (52, 149, 1600)-net in base 8, because

1 times m-reduction [i] would yield (52, 148, 1600)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 45 518622 901302 869564 615149 908504 888121 776329 469000 824492 727004 506591 099810 832578 715836 290851 722368 574244 135398 381647 077886 762693 500759 > 8¹⁴⁸ [i]