MinT - Best Known (164−99, 164, s)-Nets in Base 8

Best Known (164−99, 164, s)-Nets in Base 8

(164−99, 164, 98)-Net over F₈ — Constructive and digital

Digital (65, 164, 98)-net over F₈, using

t-expansion [i] based on digital (37, 164, 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]

(164−99, 164, 144)-Net over F₈ — Digital

Digital (65, 164, 144)-net over F₈, using

t-expansion [i] based on digital (45, 164, 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]

(164−99, 164, 2725)-Net in Base 8 — Upper bound on s

There is no (65, 164, 2726)-net in base 8, because

1 times m-reduction [i] would yield (65, 163, 2726)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1608 003425 830820 921591 806916 438366 255680 881493 838878 814471 646272 788560 267675 068821 689489 487078 091562 720328 264298 221235 897231 056649 488377 333789 424056 > 8¹⁶³ [i]