MinT - Best Known (138−83, 138, s)-Nets in Base 8

Best Known (138−83, 138, s)-Nets in Base 8

(138−83, 138, 98)-Net over F₈ — Constructive and digital

Digital (55, 138, 98)-net over F₈, using

t-expansion [i] based on digital (37, 138, 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]

(138−83, 138, 144)-Net over F₈ — Digital

Digital (55, 138, 144)-net over F₈, using

t-expansion [i] based on digital (45, 138, 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]

(138−83, 138, 2375)-Net in Base 8 — Upper bound on s

There is no (55, 138, 2376)-net in base 8, because

1 times m-reduction [i] would yield (55, 137, 2376)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5325 965432 801388 118591 365547 671310 579477 199627 343385 259179 595568 091576 775346 104763 245161 752799 706454 847325 601983 015921 186648 > 8¹³⁷ [i]