MinT - Best Known (137−83, 137, s)-Nets in Base 8

Best Known (137−83, 137, s)-Nets in Base 8

(137−83, 137, 98)-Net over F₈ — Constructive and digital

Digital (54, 137, 98)-net over F₈, using

t-expansion [i] based on digital (37, 137, 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]

(137−83, 137, 144)-Net over F₈ — Digital

Digital (54, 137, 144)-net over F₈, using

t-expansion [i] based on digital (45, 137, 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]

(137−83, 137, 2256)-Net in Base 8 — Upper bound on s

There is no (54, 137, 2257)-net in base 8, because

1 times m-reduction [i] would yield (54, 136, 2257)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 663 018321 875714 434517 143284 479082 618910 787368 202957 568357 838494 565000 090609 869735 135217 910452 116968 331536 657446 212287 832960 > 8¹³⁶ [i]