MinT - Best Known (128−83, 128, s)-Nets in Base 8

Best Known (128−83, 128, s)-Nets in Base 8

(128−83, 128, 98)-Net over F₈ — Constructive and digital

Digital (45, 128, 98)-net over F₈, using

t-expansion [i] based on digital (37, 128, 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]

(128−83, 128, 144)-Net over F₈ — Digital

Digital (45, 128, 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]

(128−83, 128, 1420)-Net in Base 8 — Upper bound on s

There is no (45, 128, 1421)-net in base 8, because

1 times m-reduction [i] would yield (45, 127, 1421)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5 015578 437994 193650 775834 807459 312122 873167 659637 659250 532190 625672 740476 409672 631321 452426 281158 942036 125345 203440 > 8¹²⁷ [i]