MinT - Best Known (51, 51+115, s)-Nets in Base 8

Best Known (51, 51+115, s)-Nets in Base 8

(51, 51+115, 98)-Net over F₈ — Constructive and digital

Digital (51, 166, 98)-net over F₈, using

t-expansion [i] based on digital (37, 166, 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]

(51, 51+115, 144)-Net over F₈ — Digital

Digital (51, 166, 144)-net over F₈, using

t-expansion [i] based on digital (45, 166, 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]

(51, 51+115, 1261)-Net in Base 8 — Upper bound on s

There is no (51, 166, 1262)-net in base 8, because

1 times m-reduction [i] would yield (51, 165, 1262)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 103866 760351 051649 374523 163688 550664 093687 925146 718067 810688 648714 713140 767399 776381 814708 322453 490142 780035 413442 782574 171352 790764 829711 975013 748028 > 8¹⁶⁵ [i]