MinT - Best Known (129−81, 129, s)-Nets in Base 8

Best Known (129−81, 129, s)-Nets in Base 8

(129−81, 129, 98)-Net over F₈ — Constructive and digital

Digital (48, 129, 98)-net over F₈, using

t-expansion [i] based on digital (37, 129, 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]

(129−81, 129, 144)-Net over F₈ — Digital

Digital (48, 129, 144)-net over F₈, using

t-expansion [i] based on digital (45, 129, 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]

(129−81, 129, 1723)-Net in Base 8 — Upper bound on s

There is no (48, 129, 1724)-net in base 8, because

1 times m-reduction [i] would yield (48, 128, 1724)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 40 262655 825257 510783 474895 099664 543442 629026 599026 167628 424026 197870 491940 438384 181393 315647 471425 949664 608884 117289 > 8¹²⁸ [i]