MinT - Best Known (132−81, 132, s)-Nets in Base 8

Best Known (132−81, 132, s)-Nets in Base 8

(132−81, 132, 98)-Net over F₈ — Constructive and digital

Digital (51, 132, 98)-net over F₈, using

t-expansion [i] based on digital (37, 132, 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]

(132−81, 132, 144)-Net over F₈ — Digital

Digital (51, 132, 144)-net over F₈, using

t-expansion [i] based on digital (45, 132, 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]

(132−81, 132, 2018)-Net in Base 8 — Upper bound on s

There is no (51, 132, 2019)-net in base 8, because

1 times m-reduction [i] would yield (51, 131, 2019)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 20536 718253 518771 893479 544969 473943 340792 184223 048894 006898 691984 452558 949689 815261 292349 826459 218457 920496 795977 808280 > 8¹³¹ [i]