MinT - Best Known (164−109, 164, s)-Nets in Base 8

Best Known (164−109, 164, s)-Nets in Base 8

(164−109, 164, 98)-Net over F₈ — Constructive and digital

Digital (55, 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]

(164−109, 164, 144)-Net over F₈ — Digital

Digital (55, 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]

(164−109, 164, 1559)-Net in Base 8 — Upper bound on s

There is no (55, 164, 1560)-net in base 8, because

1 times m-reduction [i] would yield (55, 163, 1560)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1607 866700 454667 195559 733429 471960 648427 523689 260195 451917 033268 672454 706473 602490 840608 378340 147317 610710 725100 010364 033444 295679 819304 392907 984592 > 8¹⁶³ [i]