MinT - Best Known (133−77, 133, s)-Nets in Base 8

Best Known (133−77, 133, s)-Nets in Base 8

(133−77, 133, 98)-Net over F₈ — Constructive and digital

Digital (56, 133, 98)-net over F₈, using

t-expansion [i] based on digital (37, 133, 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]

(133−77, 133, 144)-Net over F₈ — Digital

Digital (56, 133, 144)-net over F₈, using

t-expansion [i] based on digital (45, 133, 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]

(133−77, 133, 2918)-Net in Base 8 — Upper bound on s

There is no (56, 133, 2919)-net in base 8, because

1 times m-reduction [i] would yield (56, 132, 2919)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 161622 685265 147643 925793 018499 589970 452242 600993 806428 763737 052163 793092 691283 506277 849338 806960 078471 397674 994378 173175 > 8¹³² [i]